SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 92.6% → 96.2%

Time: 18.4s

Alternatives: 10

Speedup: 0.7×

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: 92.6% 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: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (* (- (tanh (/ t y)) (tanh (/ x y))) (* y z)))))
   (if (<= t_1 2e+307) t_1 (+ x (* z (- t x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + ((tanh((t / y)) - tanh((x / y))) * (y * z));
	double tmp;
	if (t_1 <= 2e+307) {
		tmp = t_1;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + ((tanh((t / y)) - tanh((x / y))) * (y * z))
    if (t_1 <= 2d+307) then
        tmp = t_1
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + ((Math.tanh((t / y)) - Math.tanh((x / y))) * (y * z));
	double tmp;
	if (t_1 <= 2e+307) {
		tmp = t_1;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + ((math.tanh((t / y)) - math.tanh((x / y))) * (y * z))
	tmp = 0
	if t_1 <= 2e+307:
		tmp = t_1
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))) * Float64(y * z)))
	tmp = 0.0
	if (t_1 <= 2e+307)
		tmp = t_1;
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + ((tanh((t / y)) - tanh((x / y))) * (y * z));
	tmp = 0.0;
	if (t_1 <= 2e+307)
		tmp = t_1;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+307], t$95$1, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.99999999999999997e307

   1. Initial program 98.3%

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

   if 1.99999999999999997e307 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 57.9%

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

   2. Taylor expanded in y around inf 94.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 2: 96.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. +-commutative 95.4%

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

   2. associate-*l* 97.3%

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

   3. fma-def 97.3%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Alternative 3: 85.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-101} \lor \neg \left(t \leq 24000000000\right):\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.6e-101) (not (<= t 24000000000.0)))
   (+ x (* z (* y (tanh (/ t y)))))
   (+ x (* (* y z) (- (/ t y) (tanh (/ x y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.6e-101) || !(t <= 24000000000.0)) {
		tmp = x + (z * (y * tanh((t / y))));
	} else {
		tmp = x + ((y * z) * ((t / y) - tanh((x / y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.6d-101)) .or. (.not. (t <= 24000000000.0d0))) then
        tmp = x + (z * (y * tanh((t / y))))
    else
        tmp = x + ((y * z) * ((t / y) - tanh((x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.6e-101) || !(t <= 24000000000.0)) {
		tmp = x + (z * (y * Math.tanh((t / y))));
	} else {
		tmp = x + ((y * z) * ((t / y) - Math.tanh((x / y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.6e-101) or not (t <= 24000000000.0):
		tmp = x + (z * (y * math.tanh((t / y))))
	else:
		tmp = x + ((y * z) * ((t / y) - math.tanh((x / y))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.6e-101) || !(t <= 24000000000.0))
		tmp = Float64(x + Float64(z * Float64(y * tanh(Float64(t / y)))));
	else
		tmp = Float64(x + Float64(Float64(y * z) * Float64(Float64(t / y) - tanh(Float64(x / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.6e-101) || ~((t <= 24000000000.0)))
		tmp = x + (z * (y * tanh((t / y))));
	else
		tmp = x + ((y * z) * ((t / y) - tanh((x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.6e-101], N[Not[LessEqual[t, 24000000000.0]], $MachinePrecision]], N[(x + N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.59999999999999989e-101 or 2.4e10 < 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. Taylor expanded in x around 0 10.2%

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

      1. associate-*r* 10.1%

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

      2. *-commutative 10.1%

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

      3. associate-/r* 10.1%

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

      4. div-sub 10.1%

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

      5. rec-exp 10.1%

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

      6. rec-exp 10.1%

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

      7. tanh-def-a 86.0%

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

      8. associate-*l* 89.0%

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

   4. Simplified 89.0%

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

   if -1.59999999999999989e-101 < t < 2.4e10

   1. Initial program 94.3%

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

   2. Taylor expanded in t around 0 87.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-101} \lor \neg \left(t \leq 24000000000\right):\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\\ \end{array} \]

Alternative 4: 85.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-101} \lor \neg \left(t \leq 3500000000\right):\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(z \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.45e-101) (not (<= t 3500000000.0)))
   (+ x (* z (* y (tanh (/ t y)))))
   (- x (* y (* z (tanh (/ x y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.45e-101) || !(t <= 3500000000.0)) {
		tmp = x + (z * (y * tanh((t / y))));
	} else {
		tmp = x - (y * (z * tanh((x / y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.45d-101)) .or. (.not. (t <= 3500000000.0d0))) then
        tmp = x + (z * (y * tanh((t / y))))
    else
        tmp = x - (y * (z * tanh((x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.45e-101) || !(t <= 3500000000.0)) {
		tmp = x + (z * (y * Math.tanh((t / y))));
	} else {
		tmp = x - (y * (z * Math.tanh((x / y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.45e-101) or not (t <= 3500000000.0):
		tmp = x + (z * (y * math.tanh((t / y))))
	else:
		tmp = x - (y * (z * math.tanh((x / y))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.45e-101) || !(t <= 3500000000.0))
		tmp = Float64(x + Float64(z * Float64(y * tanh(Float64(t / y)))));
	else
		tmp = Float64(x - Float64(y * Float64(z * tanh(Float64(x / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.45e-101) || ~((t <= 3500000000.0)))
		tmp = x + (z * (y * tanh((t / y))));
	else
		tmp = x - (y * (z * tanh((x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.45e-101], N[Not[LessEqual[t, 3500000000.0]], $MachinePrecision]], N[(x + N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(z * N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.45e-101 or 3.5e9 < 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. Taylor expanded in x around 0 10.2%

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

      1. associate-*r* 10.1%

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

      2. *-commutative 10.1%

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

      3. associate-/r* 10.1%

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

      4. div-sub 10.1%

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

      5. rec-exp 10.1%

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

      6. rec-exp 10.1%

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

      7. tanh-def-a 86.0%

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

      8. associate-*l* 89.0%

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

   4. Simplified 89.0%

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

   if -1.45e-101 < t < 3.5e9

   1. Initial program 94.3%

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

      1. flip-- 74.1%

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

      2. associate-*r/ 74.1%

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

      3. pow2 74.1%

         \[\leadsto x + \frac{\left(y \cdot z\right) \cdot \left(\color{blue}{{\tanh \left(\frac{t}{y}\right)}^{2}} - \tanh \left(\frac{x}{y}\right) \cdot \tanh \left(\frac{x}{y}\right)\right)}{\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)} \]

      4. pow2 74.1%

         \[\leadsto x + \frac{\left(y \cdot z\right) \cdot \left({\tanh \left(\frac{t}{y}\right)}^{2} - \color{blue}{{\tanh \left(\frac{x}{y}\right)}^{2}}\right)}{\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)} \]

   3. Applied egg-rr 74.1%

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

      1. frac-2neg 74.1%

         \[\leadsto x + \color{blue}{\frac{-\left(y \cdot z\right) \cdot \left({\tanh \left(\frac{t}{y}\right)}^{2} - {\tanh \left(\frac{x}{y}\right)}^{2}\right)}{-\left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)}} \]

      2. neg-sub0 74.1%

         \[\leadsto x + \frac{\color{blue}{0 - \left(y \cdot z\right) \cdot \left({\tanh \left(\frac{t}{y}\right)}^{2} - {\tanh \left(\frac{x}{y}\right)}^{2}\right)}}{-\left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)} \]

      3. div-sub 74.1%

         \[\leadsto x + \color{blue}{\left(\frac{0}{-\left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)} - \frac{\left(y \cdot z\right) \cdot \left({\tanh \left(\frac{t}{y}\right)}^{2} - {\tanh \left(\frac{x}{y}\right)}^{2}\right)}{-\left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)}\right)} \]

      4. neg-mul-1 74.1%

         \[\leadsto x + \left(\frac{0}{-\left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)} - \frac{\left(y \cdot z\right) \cdot \left({\tanh \left(\frac{t}{y}\right)}^{2} - {\tanh \left(\frac{x}{y}\right)}^{2}\right)}{\color{blue}{-1 \cdot \left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)}}\right) \]

      5. metadata-eval 74.1%

         \[\leadsto x + \left(\frac{0}{-\left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)} - \frac{\left(y \cdot z\right) \cdot \left({\tanh \left(\frac{t}{y}\right)}^{2} - {\tanh \left(\frac{x}{y}\right)}^{2}\right)}{\color{blue}{\left(-1\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)}\right) \]

      6. times-frac 74.1%

         \[\leadsto x + \left(\frac{0}{-\left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)} - \color{blue}{\frac{y \cdot z}{-1} \cdot \frac{{\tanh \left(\frac{t}{y}\right)}^{2} - {\tanh \left(\frac{x}{y}\right)}^{2}}{\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)}}\right) \]

      7. metadata-eval 74.1%

         \[\leadsto x + \left(\frac{0}{-\left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)} - \frac{y \cdot z}{\color{blue}{-1}} \cdot \frac{{\tanh \left(\frac{t}{y}\right)}^{2} - {\tanh \left(\frac{x}{y}\right)}^{2}}{\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)}\right) \]

   5. Applied egg-rr 80.6%

      \[\leadsto x + \color{blue}{\left(\frac{0}{-\left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)} - \frac{y \cdot z}{-1} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. div0 94.3%

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

      2. neg-sub0 94.3%

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

      3. distribute-rgt-neg-in 94.3%

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

      4. associate-/l* 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in t around 0 17.0%

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

      1. associate-*r* 17.0%

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

      2. neg-mul-1 17.0%

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

      3. associate-/r* 17.0%

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

      4. rec-exp 17.0%

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

      5. div-sub 17.0%

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

      6. rec-exp 17.0%

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

      7. tanh-def-a 85.3%

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

   10. Simplified 85.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-101} \lor \neg \left(t \leq 3500000000\right):\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(z \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \end{array} \]

Alternative 5: 83.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+120}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.22e+120) (+ x (* z (* y (tanh (/ t y))))) (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.22e+120) {
		tmp = x + (z * (y * tanh((t / y))));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.22d+120) then
        tmp = x + (z * (y * tanh((t / y))))
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.22e+120) {
		tmp = x + (z * (y * Math.tanh((t / y))));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.22e+120:
		tmp = x + (z * (y * math.tanh((t / y))))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.22e+120)
		tmp = Float64(x + Float64(z * Float64(y * tanh(Float64(t / y)))));
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.22e+120)
		tmp = x + (z * (y * tanh((t / y))));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.22e+120], N[(x + N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.22e120

   1. Initial program 97.3%

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

   2. Taylor expanded in x around 0 21.4%

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

      1. associate-*r* 21.3%

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

      2. *-commutative 21.3%

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

      3. associate-/r* 21.3%

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

      4. div-sub 21.3%

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

      5. rec-exp 21.3%

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

      6. rec-exp 21.3%

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

      7. tanh-def-a 78.3%

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

      8. associate-*l* 79.2%

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

   4. Simplified 79.2%

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

   if 1.22e120 < y

   1. Initial program 84.1%

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

   2. Taylor expanded in y around inf 91.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+120}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 6: 63.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+188}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1350000000.0) x (if (<= y 1.05e+188) (- x (* z x)) (+ x (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1350000000.0) {
		tmp = x;
	} else if (y <= 1.05e+188) {
		tmp = x - (z * x);
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1350000000.0d0) then
        tmp = x
    else if (y <= 1.05d+188) then
        tmp = x - (z * x)
    else
        tmp = x + (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1350000000.0) {
		tmp = x;
	} else if (y <= 1.05e+188) {
		tmp = x - (z * x);
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1350000000.0:
		tmp = x
	elif y <= 1.05e+188:
		tmp = x - (z * x)
	else:
		tmp = x + (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1350000000.0)
		tmp = x;
	elseif (y <= 1.05e+188)
		tmp = Float64(x - Float64(z * x));
	else
		tmp = Float64(x + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1350000000.0)
		tmp = x;
	elseif (y <= 1.05e+188)
		tmp = x - (z * x);
	else
		tmp = x + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1350000000.0], x, If[LessEqual[y, 1.05e+188], N[(x - N[(z * x), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.35e9

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-*l* 98.1%

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

      3. fma-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in y around 0 64.5%

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

   if 1.35e9 < y < 1.04999999999999993e188

   1. Initial program 94.9%

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

   2. Taylor expanded in x around 0 80.6%

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

   3. Taylor expanded in t around 0 66.0%

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

   4. Taylor expanded in t around 0 58.5%

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

      1. mul-1-neg 58.5%

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

      2. distribute-frac-neg 58.5%

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

   6. Simplified 58.5%

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

   7. Taylor expanded in x around 0 61.1%

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

      1. mul-1-neg 61.1%

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

      2. distribute-rgt-in 61.1%

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

      3. *-lft-identity 61.1%

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

      4. cancel-sign-sub-inv 61.1%

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

      5. *-commutative 61.1%

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

   9. Simplified 61.1%

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

   if 1.04999999999999993e188 < y

   1. Initial program 77.8%

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

   2. Taylor expanded in x around 0 31.0%

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

      1. associate-*r* 30.1%

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

      2. *-commutative 30.1%

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

      3. associate-/r* 30.1%

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

      4. div-sub 30.1%

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

      5. rec-exp 30.1%

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

      6. rec-exp 30.1%

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

      7. tanh-def-a 66.2%

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

      8. associate-*l* 82.9%

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

   4. Simplified 82.9%

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

   5. Taylor expanded in y around inf 78.2%

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

      1. +-commutative 78.2%

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

   7. Simplified 78.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+188}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \]

Alternative 7: 68.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.25e-35) x (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.25e-35) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.25d-35) then
        tmp = x
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.25e-35) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.25e-35:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.25e-35)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.25e-35)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.25e-35], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.24999999999999991e-35

   1. Initial program 96.9%

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

      1. +-commutative 96.9%

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

      2. associate-*l* 98.0%

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

      3. fma-def 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in y around 0 65.3%

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

   if 1.24999999999999991e-35 < y

   1. Initial program 91.5%

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

   2. Taylor expanded in y around inf 73.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 8: 59.8% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-227}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.85e-117) x (if (<= x 2.5e-227) (* z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.85e-117) {
		tmp = x;
	} else if (x <= 2.5e-227) {
		tmp = z * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.85d-117)) then
        tmp = x
    else if (x <= 2.5d-227) then
        tmp = z * t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.85e-117) {
		tmp = x;
	} else if (x <= 2.5e-227) {
		tmp = z * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.85e-117:
		tmp = x
	elif x <= 2.5e-227:
		tmp = z * t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.85e-117)
		tmp = x;
	elseif (x <= 2.5e-227)
		tmp = Float64(z * t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.85e-117)
		tmp = x;
	elseif (x <= 2.5e-227)
		tmp = z * t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.85e-117], x, If[LessEqual[x, 2.5e-227], N[(z * t), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.85e-117 or 2.4999999999999998e-227 < x

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. associate-*l* 99.9%

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

      3. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 69.1%

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

   if -2.85e-117 < x < 2.4999999999999998e-227

   1. Initial program 90.3%

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

   2. Taylor expanded in x around 0 9.6%

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

      1. associate-*r* 9.4%

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

      2. *-commutative 9.4%

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

      3. associate-/r* 9.4%

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

      4. div-sub 9.4%

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

      5. rec-exp 9.4%

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

      6. rec-exp 9.4%

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

      7. tanh-def-a 74.2%

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

      8. associate-*l* 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in y around inf 49.5%

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

      1. +-commutative 49.5%

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

   7. Simplified 49.5%

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

   8. Taylor expanded in t around inf 40.2%

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

      1. *-commutative 40.2%

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

   10. Simplified 40.2%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-227}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 64.9% accurate, 30.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 4.4e+67) x (+ x (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.4e+67) {
		tmp = x;
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 4.4d+67) then
        tmp = x
    else
        tmp = x + (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.4e+67) {
		tmp = x;
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.4e+67:
		tmp = x
	else:
		tmp = x + (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.4e+67)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.4e+67)
		tmp = x;
	else
		tmp = x + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.4e+67], x, N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4e67

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

      2. associate-*l* 98.2%

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

      3. fma-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in y around 0 62.9%

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

   if 4.4e67 < y

   1. Initial program 88.2%

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

   2. Taylor expanded in x around 0 31.6%

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

      1. associate-*r* 31.1%

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

      2. *-commutative 31.1%

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

      3. associate-/r* 31.1%

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

      4. div-sub 31.1%

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

      5. rec-exp 31.2%

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

      6. rec-exp 31.2%

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

      7. tanh-def-a 66.2%

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

      8. associate-*l* 72.2%

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

   4. Simplified 72.2%

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

   5. Taylor expanded in y around inf 62.8%

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

      1. +-commutative 62.8%

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

   7. Simplified 62.8%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \]

Alternative 10: 59.2% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. +-commutative 95.4%

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

   2. associate-*l* 97.3%

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

   3. fma-def 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in y around 0 57.9%

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

5. Final simplification 57.9%

   \[\leadsto x \]

Developer target: 96.9% 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 2023297 
(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))))))