SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.8% → 98.2%

Time: 11.7s

Alternatives: 13

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: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+179}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.02e+179)
   (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))
   (fma z (- t x) x)))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.02e+179) {
		tmp = x + (y * (z * (tanh((t / y)) - tanh((x / y)))));
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.02e+179)
		tmp = Float64(x + Float64(y * Float64(z * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))))));
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 1.02e+179], 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], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0199999999999999e179

   1. Initial program 95.1%

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

      1. associate-*l* 96.3%

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

   3. Simplified 96.3%

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

   if 1.0199999999999999e179 < y

   1. Initial program 80.6%

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

      1. associate-*l* 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+179}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 2: 97.9% 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 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* 97.1%

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

   4. fma-def 97.1%

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

3. Simplified 97.1%

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

4. Final simplification 97.1%

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

Alternative 3: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-126} \lor \neg \left(t \leq 1.4 \cdot 10^{-66}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.95e-126) (not (<= t 1.4e-66)))
   (fma z (* y (tanh (/ t y))) x)
   (+ x (* y (* z (- (/ t y) (tanh (/ x y))))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.95e-126) || !(t <= 1.4e-66)) {
		tmp = fma(z, (y * tanh((t / y))), x);
	} else {
		tmp = x + (y * (z * ((t / y) - tanh((x / y)))));
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.95e-126) || !(t <= 1.4e-66))
		tmp = fma(z, Float64(y * tanh(Float64(t / y))), x);
	else
		tmp = Float64(x + Float64(y * Float64(z * 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, -1.95e-126], N[Not[LessEqual[t, 1.4e-66]], $MachinePrecision]], N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(y * N[(z * 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 < -1.9499999999999999e-126 or 1.4e-66 < t

   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. Step-by-step derivation

      1. +-commutative 97.7%

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

      2. *-commutative 97.7%

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

      3. associate-*l* 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 13.1%

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

      1. associate-/r* 13.1%

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

      2. div-sub 13.1%

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

      3. rec-exp 13.1%

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

      4. rec-exp 13.1%

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

      5. tanh-def-a 93.5%

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

   6. Simplified 93.5%

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

   if -1.9499999999999999e-126 < t < 1.4e-66

   1. Initial program 86.7%

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

      1. associate-*l* 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in t around 0 88.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-126} \lor \neg \left(t \leq 1.4 \cdot 10^{-66}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 4: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-54} \lor \neg \left(t \leq 2.9 \cdot 10^{-63}\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.7e-54) (not (<= t 2.9e-63)))
   (+ x (* (tanh (/ t y)) (* z y)))
   (+ x (* y (* z (- (/ t y) (tanh (/ x y))))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.7e-54) || !(t <= 2.9e-63)) {
		tmp = x + (tanh((t / y)) * (z * y));
	} else {
		tmp = x + (y * (z * ((t / y) - tanh((x / y)))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.7d-54)) .or. (.not. (t <= 2.9d-63))) then
        tmp = x + (tanh((t / y)) * (z * y))
    else
        tmp = x + (y * (z * ((t / y) - tanh((x / y)))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.7e-54) || !(t <= 2.9e-63)) {
		tmp = x + (Math.tanh((t / y)) * (z * y));
	} else {
		tmp = x + (y * (z * ((t / y) - Math.tanh((x / y)))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if (t <= -2.7e-54) or not (t <= 2.9e-63):
		tmp = x + (math.tanh((t / y)) * (z * y))
	else:
		tmp = x + (y * (z * ((t / y) - math.tanh((x / y)))))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.7e-54) || !(t <= 2.9e-63))
		tmp = Float64(x + Float64(tanh(Float64(t / y)) * Float64(z * y)));
	else
		tmp = Float64(x + Float64(y * Float64(z * Float64(Float64(t / y) - tanh(Float64(x / y))))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.7e-54) || ~((t <= 2.9e-63)))
		tmp = x + (tanh((t / y)) * (z * y));
	else
		tmp = x + (y * (z * ((t / y) - tanh((x / y)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.7e-54], N[Not[LessEqual[t, 2.9e-63]], $MachinePrecision]], N[(x + N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z * 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 < -2.70000000000000026e-54 or 2.89999999999999975e-63 < t

   1. Initial program 98.2%

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

      1. associate-*l* 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in x around 0 13.1%

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

      1. *-commutative 13.1%

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

      2. associate-*r* 13.0%

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

      3. associate-/r* 13.0%

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

      4. div-sub 13.0%

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

      5. rec-exp 13.0%

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

      6. rec-exp 13.0%

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

   6. Simplified 92.4%

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

   if -2.70000000000000026e-54 < t < 2.89999999999999975e-63

   1. Initial program 86.9%

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

      1. associate-*l* 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in t around 0 87.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-54} \lor \neg \left(t \leq 2.9 \cdot 10^{-63}\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 5: 81.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-127} \lor \neg \left(t \leq 1.35 \cdot 10^{-68}\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z \cdot t - z \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.9e-127) (not (<= t 1.35e-68)))
   (+ x (* (tanh (/ t y)) (* z y)))
   (+ x (- (* z t) (* z x)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.9e-127) || !(t <= 1.35e-68)) {
		tmp = x + (tanh((t / y)) * (z * y));
	} else {
		tmp = x + ((z * t) - (z * 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 ((t <= (-2.9d-127)) .or. (.not. (t <= 1.35d-68))) then
        tmp = x + (tanh((t / y)) * (z * y))
    else
        tmp = x + ((z * t) - (z * 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 ((t <= -2.9e-127) || !(t <= 1.35e-68)) {
		tmp = x + (Math.tanh((t / y)) * (z * y));
	} else {
		tmp = x + ((z * t) - (z * x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if (t <= -2.9e-127) or not (t <= 1.35e-68):
		tmp = x + (math.tanh((t / y)) * (z * y))
	else:
		tmp = x + ((z * t) - (z * x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.9e-127) || !(t <= 1.35e-68))
		tmp = Float64(x + Float64(tanh(Float64(t / y)) * Float64(z * y)));
	else
		tmp = Float64(x + Float64(Float64(z * t) - Float64(z * x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.9e-127) || ~((t <= 1.35e-68)))
		tmp = x + (tanh((t / y)) * (z * y));
	else
		tmp = x + ((z * t) - (z * x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.9e-127], N[Not[LessEqual[t, 1.35e-68]], $MachinePrecision]], N[(x + N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z * t), $MachinePrecision] - N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9e-127 or 1.3500000000000001e-68 < t

   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. 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 13.1%

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

      1. *-commutative 13.1%

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

      2. associate-*r* 12.9%

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

      3. associate-/r* 12.9%

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

      4. div-sub 12.9%

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

      5. rec-exp 13.0%

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

      6. rec-exp 13.0%

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

   6. Simplified 91.1%

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

   if -2.9e-127 < t < 1.3500000000000001e-68

   1. Initial program 86.7%

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

      1. +-commutative 86.7%

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

      2. *-commutative 86.7%

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

      3. associate-*l* 92.0%

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

      4. fma-def 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in y around inf 79.8%

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

   5. Taylor expanded in t around 0 79.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-127} \lor \neg \left(t \leq 1.35 \cdot 10^{-68}\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z \cdot t - z \cdot x\right)\\ \end{array} \]

Alternative 6: 75.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.46 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t) :precision binary64 (if (<= y 1.46e-93) x (fma z (- t x) x)))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.46e-93) {
		tmp = x;
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.46e-93)
		tmp = x;
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 1.46e-93], x, N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45999999999999999e-93

   1. Initial program 95.1%

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

      1. associate-*l* 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in x around inf 62.3%

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

   if 1.45999999999999999e-93 < y

   1. Initial program 90.6%

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

      1. associate-*l* 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around inf 81.1%

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

      1. +-commutative 81.1%

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

      2. fma-def 81.1%

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

   6. Simplified 81.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.46 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 7: 67.9% accurate, 13.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_1 := x \cdot \left(1 - z\right)\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+78} \lor \neg \left(y \leq 4.1 \cdot 10^{+86}\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- 1.0 z))))
   (if (<= y 6.8e-109)
     x
     (if (<= y 4.6e+21)
       t_1
       (if (<= y 8.2e+58)
         x
         (if (or (<= y 2.3e+78) (not (<= y 4.1e+86))) (+ x (* z t)) t_1))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double t_1 = x * (1.0 - z);
	double tmp;
	if (y <= 6.8e-109) {
		tmp = x;
	} else if (y <= 4.6e+21) {
		tmp = t_1;
	} else if (y <= 8.2e+58) {
		tmp = x;
	} else if ((y <= 2.3e+78) || !(y <= 4.1e+86)) {
		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 * (1.0d0 - z)
    if (y <= 6.8d-109) then
        tmp = x
    else if (y <= 4.6d+21) then
        tmp = t_1
    else if (y <= 8.2d+58) then
        tmp = x
    else if ((y <= 2.3d+78) .or. (.not. (y <= 4.1d+86))) 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 * (1.0 - z);
	double tmp;
	if (y <= 6.8e-109) {
		tmp = x;
	} else if (y <= 4.6e+21) {
		tmp = t_1;
	} else if (y <= 8.2e+58) {
		tmp = x;
	} else if ((y <= 2.3e+78) || !(y <= 4.1e+86)) {
		tmp = x + (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	t_1 = x * (1.0 - z)
	tmp = 0
	if y <= 6.8e-109:
		tmp = x
	elif y <= 4.6e+21:
		tmp = t_1
	elif y <= 8.2e+58:
		tmp = x
	elif (y <= 2.3e+78) or not (y <= 4.1e+86):
		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(1.0 - z))
	tmp = 0.0
	if (y <= 6.8e-109)
		tmp = x;
	elseif (y <= 4.6e+21)
		tmp = t_1;
	elseif (y <= 8.2e+58)
		tmp = x;
	elseif ((y <= 2.3e+78) || !(y <= 4.1e+86))
		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 * (1.0 - z);
	tmp = 0.0;
	if (y <= 6.8e-109)
		tmp = x;
	elseif (y <= 4.6e+21)
		tmp = t_1;
	elseif (y <= 8.2e+58)
		tmp = x;
	elseif ((y <= 2.3e+78) || ~((y <= 4.1e+86)))
		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[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.8e-109], x, If[LessEqual[y, 4.6e+21], t$95$1, If[LessEqual[y, 8.2e+58], x, If[Or[LessEqual[y, 2.3e+78], N[Not[LessEqual[y, 4.1e+86]], $MachinePrecision]], N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.80000000000000023e-109 or 4.6e21 < y < 8.2e58

   1. Initial program 95.3%

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

      1. associate-*l* 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in x around inf 63.4%

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

   if 6.80000000000000023e-109 < y < 4.6e21 or 2.3000000000000002e78 < y < 4.0999999999999999e86

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 69.5%

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

   5. Taylor expanded in t around 0 72.4%

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

      1. associate-*r* 72.4%

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

      2. *-lft-identity 72.4%

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

      3. distribute-rgt-in 72.4%

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

      4. +-commutative 72.4%

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

      5. mul-1-neg 72.4%

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

      6. unsub-neg 72.4%

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

   7. Simplified 72.4%

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

   if 8.2e58 < y < 2.3000000000000002e78 or 4.0999999999999999e86 < y

   1. Initial program 83.0%

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

      1. associate-*l* 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in x around 0 31.5%

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

      1. *-commutative 31.5%

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

      2. associate-*r* 30.6%

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

      3. associate-/r* 30.6%

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

      4. div-sub 30.6%

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

      5. rec-exp 30.6%

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

      6. rec-exp 30.6%

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

   6. Simplified 69.2%

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

   7. Taylor expanded in y around inf 69.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+78} \lor \neg \left(y \leq 4.1 \cdot 10^{+86}\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 8: 60.7% accurate, 18.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-61}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-207}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-190}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.6e-32)
   x
   (if (<= x -4.8e-61)
     (* z (- x))
     (if (<= x -5e-207) x (if (<= x 4.5e-190) (* z t) x)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.6e-32) {
		tmp = x;
	} else if (x <= -4.8e-61) {
		tmp = z * -x;
	} else if (x <= -5e-207) {
		tmp = x;
	} else if (x <= 4.5e-190) {
		tmp = z * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-6.6d-32)) then
        tmp = x
    else if (x <= (-4.8d-61)) then
        tmp = z * -x
    else if (x <= (-5d-207)) then
        tmp = x
    else if (x <= 4.5d-190) then
        tmp = z * t
    else
        tmp = x
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.6e-32) {
		tmp = x;
	} else if (x <= -4.8e-61) {
		tmp = z * -x;
	} else if (x <= -5e-207) {
		tmp = x;
	} else if (x <= 4.5e-190) {
		tmp = z * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if x <= -6.6e-32:
		tmp = x
	elif x <= -4.8e-61:
		tmp = z * -x
	elif x <= -5e-207:
		tmp = x
	elif x <= 4.5e-190:
		tmp = z * t
	else:
		tmp = x
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.6e-32)
		tmp = x;
	elseif (x <= -4.8e-61)
		tmp = Float64(z * Float64(-x));
	elseif (x <= -5e-207)
		tmp = x;
	elseif (x <= 4.5e-190)
		tmp = Float64(z * t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -6.6e-32)
		tmp = x;
	elseif (x <= -4.8e-61)
		tmp = z * -x;
	elseif (x <= -5e-207)
		tmp = x;
	elseif (x <= 4.5e-190)
		tmp = z * t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[x, -6.6e-32], x, If[LessEqual[x, -4.8e-61], N[(z * (-x)), $MachinePrecision], If[LessEqual[x, -5e-207], x, If[LessEqual[x, 4.5e-190], N[(z * t), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.60000000000000051e-32 or -4.8000000000000002e-61 < x < -5.00000000000000014e-207 or 4.50000000000000021e-190 < x

   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. Step-by-step derivation

      1. associate-*l* 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in x around inf 69.5%

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

   if -6.60000000000000051e-32 < x < -4.8000000000000002e-61

   1. Initial program 89.0%

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

      1. +-commutative 89.0%

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

      2. *-commutative 89.0%

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

      3. associate-*l* 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 68.3%

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

   5. Taylor expanded in z around inf 68.3%

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

   6. Taylor expanded in t around 0 57.2%

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

      1. associate-*r* 57.2%

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

      2. mul-1-neg 57.2%

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

   8. Simplified 57.2%

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

   if -5.00000000000000014e-207 < x < 4.50000000000000021e-190

   1. Initial program 83.8%

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

      1. +-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 89.4%

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

      4. fma-def 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in y around inf 59.4%

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

   5. Taylor expanded in t around inf 47.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-61}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-207}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-190}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 61.4% accurate, 19.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+208} \lor \neg \left(y \leq 1.7 \cdot 10^{+274}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.8e-109)
   x
   (if (or (<= y 9.8e+208) (not (<= y 1.7e+274))) (* x (- 1.0 z)) (* z t))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.8e-109) {
		tmp = x;
	} else if ((y <= 9.8e+208) || !(y <= 1.7e+274)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = 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 <= 6.8d-109) then
        tmp = x
    else if ((y <= 9.8d+208) .or. (.not. (y <= 1.7d+274))) then
        tmp = x * (1.0d0 - z)
    else
        tmp = 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 <= 6.8e-109) {
		tmp = x;
	} else if ((y <= 9.8e+208) || !(y <= 1.7e+274)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 6.8e-109:
		tmp = x
	elif (y <= 9.8e+208) or not (y <= 1.7e+274):
		tmp = x * (1.0 - z)
	else:
		tmp = z * t
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.8e-109)
		tmp = x;
	elseif ((y <= 9.8e+208) || !(y <= 1.7e+274))
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 6.8e-109)
		tmp = x;
	elseif ((y <= 9.8e+208) || ~((y <= 1.7e+274)))
		tmp = x * (1.0 - z);
	else
		tmp = 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, 6.8e-109], x, If[Or[LessEqual[y, 9.8e+208], N[Not[LessEqual[y, 1.7e+274]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.80000000000000023e-109

   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* 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in x around inf 62.3%

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

   if 6.80000000000000023e-109 < y < 9.79999999999999918e208 or 1.7e274 < y

   1. Initial program 92.3%

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

      1. +-commutative 92.3%

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

      2. *-commutative 92.3%

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

      3. associate-*l* 94.8%

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

      4. fma-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in y around inf 77.0%

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

   5. Taylor expanded in t around 0 63.9%

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

      1. associate-*r* 63.9%

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

      2. *-lft-identity 63.9%

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

      3. distribute-rgt-in 63.9%

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

      4. +-commutative 63.9%

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

      5. mul-1-neg 63.9%

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

      6. unsub-neg 63.9%

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

   7. Simplified 63.9%

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

   if 9.79999999999999918e208 < y < 1.7e274

   1. Initial program 79.5%

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

      1. +-commutative 79.5%

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

      2. *-commutative 79.5%

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

      3. associate-*l* 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   5. Taylor expanded in t around inf 56.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+208} \lor \neg \left(y \leq 1.7 \cdot 10^{+274}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 10: 64.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+63} \lor \neg \left(z \leq 3.4 \cdot 10^{+166}\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4e+63) (not (<= z 3.4e+166))) (* z (- t x)) x))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4e+63) || !(z <= 3.4e+166)) {
		tmp = z * (t - x);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-4d+63)) .or. (.not. (z <= 3.4d+166))) then
        tmp = z * (t - x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4e+63) || !(z <= 3.4e+166)) {
		tmp = z * (t - x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if (z <= -4e+63) or not (z <= 3.4e+166):
		tmp = z * (t - x)
	else:
		tmp = x
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4e+63) || !(z <= 3.4e+166))
		tmp = Float64(z * Float64(t - x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4e+63) || ~((z <= 3.4e+166)))
		tmp = z * (t - x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4e+63], N[Not[LessEqual[z, 3.4e+166]], $MachinePrecision]], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -4.00000000000000023e63 or 3.4e166 < z

   1. Initial program 85.3%

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

      1. +-commutative 85.3%

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

      2. *-commutative 85.3%

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

      3. associate-*l* 94.1%

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

      4. fma-def 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in y around inf 60.4%

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

   5. Taylor expanded in z around inf 60.4%

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

   if -4.00000000000000023e63 < z < 3.4e166

   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. Step-by-step derivation

      1. associate-*l* 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in x around inf 67.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+63} \lor \neg \left(z \leq 3.4 \cdot 10^{+166}\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 75.7% accurate, 23.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.4e-93) x (+ x (* z (- t x)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.4e-93) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.4d-93) then
        tmp = x
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.4e-93) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 1.4e-93:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.4e-93)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.4e-93)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 1.4e-93], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.39999999999999999e-93

   1. Initial program 95.1%

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

      1. associate-*l* 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in x around inf 62.3%

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

   if 1.39999999999999999e-93 < y

   1. Initial program 90.6%

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

      1. +-commutative 90.6%

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

      2. *-commutative 90.6%

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

      3. associate-*l* 95.2%

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

      4. fma-def 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in y around inf 81.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 12: 61.7% accurate, 29.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-189}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x -7.8e-208) x (if (<= x 7e-189) (* z t) x)))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.8e-208) {
		tmp = x;
	} else if (x <= 7e-189) {
		tmp = z * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-7.8d-208)) then
        tmp = x
    else if (x <= 7d-189) then
        tmp = z * t
    else
        tmp = x
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.8e-208) {
		tmp = x;
	} else if (x <= 7e-189) {
		tmp = z * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if x <= -7.8e-208:
		tmp = x
	elif x <= 7e-189:
		tmp = z * t
	else:
		tmp = x
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7.8e-208)
		tmp = x;
	elseif (x <= 7e-189)
		tmp = Float64(z * t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -7.8e-208)
		tmp = x;
	elseif (x <= 7e-189)
		tmp = z * t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[x, -7.8e-208], x, If[LessEqual[x, 7e-189], N[(z * t), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.80000000000000007e-208 or 7.0000000000000003e-189 < x

   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. associate-*l* 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in x around inf 66.5%

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

   if -7.80000000000000007e-208 < x < 7.0000000000000003e-189

   1. Initial program 83.8%

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

      1. +-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 89.4%

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

      4. fma-def 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in y around inf 59.4%

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

   5. Taylor expanded in t around inf 47.4%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-189}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13: 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 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. associate-*l* 95.5%

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

3. Simplified 95.5%

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

4. Taylor expanded in x around inf 57.1%

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

5. Final simplification 57.1%

   \[\leadsto x \]

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 2023222 
(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))))))