SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.7% → 99.0%

Time: 16.2s

Alternatives: 10

Speedup: 0.5×

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: 93.7% 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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{x}{y}\right)\\ t_2 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;x - \left(y \cdot z\right) \cdot \left(t\_1 - t\_2\right) \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(t\_2 - t\_1\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ x y))) (t_2 (tanh (/ t y))))
   (if (<= (- x (* (* y z) (- t_1 t_2))) 1e+308)
     (fma (* y (- t_2 t_1)) z x)
     (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((x / y));
	double t_2 = tanh((t / y));
	double tmp;
	if ((x - ((y * z) * (t_1 - t_2))) <= 1e+308) {
		tmp = fma((y * (t_2 - t_1)), z, x);
	} else {
		tmp = z * (t - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = tanh(Float64(x / y))
	t_2 = tanh(Float64(t / y))
	tmp = 0.0
	if (Float64(x - Float64(Float64(y * z) * Float64(t_1 - t_2))) <= 1e+308)
		tmp = fma(Float64(y * Float64(t_2 - t_1)), z, x);
	else
		tmp = Float64(z * Float64(t - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x - N[(N[(y * z), $MachinePrecision] * N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+308], N[(N[(y * N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], N[(z * N[(t - x), $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))))) < 1e308

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. fma-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

   8. Simplified 100.0%

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

   if 1e308 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 40.6%

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

      1. cancel-sign-sub 40.6%

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

      2. distribute-lft-neg-out 40.6%

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

      3. distribute-rgt-neg-in 40.6%

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

      4. neg-sub0 40.6%

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

      5. associate-+l- 40.6%

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

      6. neg-sub0 40.6%

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

      7. +-commutative 40.6%

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

      8. sub-neg 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in t around 0 37.6%

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

      1. associate-*r* 0.0%

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

      2. associate-/r* 0.0%

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

      3. rec-exp 0.0%

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

      4. div-sub 0.0%

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

      5. rec-exp 0.0%

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

      6. tanh-def-a 34.6%

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

   7. Simplified 34.6%

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

   8. Taylor expanded in z around inf 37.2%

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

      1. associate-/l* 37.2%

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

      2. rec-exp 37.2%

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

      3. rec-exp 37.2%

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

      4. tanh-def-a 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in y around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;x - \left(y \cdot z\right) \cdot t\_1 \leq 10^{+308}:\\ \;\;\;\;x - z \cdot \left(y \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (tanh (/ x y)) (tanh (/ t y)))))
   (if (<= (- x (* (* y z) t_1)) 1e+308) (- x (* z (* y t_1))) (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((x / y)) - tanh((t / y));
	double tmp;
	if ((x - ((y * z) * t_1)) <= 1e+308) {
		tmp = x - (z * (y * t_1));
	} else {
		tmp = 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 = tanh((x / y)) - tanh((t / y))
    if ((x - ((y * z) * t_1)) <= 1d+308) then
        tmp = x - (z * (y * t_1))
    else
        tmp = 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 = Math.tanh((x / y)) - Math.tanh((t / y));
	double tmp;
	if ((x - ((y * z) * t_1)) <= 1e+308) {
		tmp = x - (z * (y * t_1));
	} else {
		tmp = z * (t - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.tanh((x / y)) - math.tanh((t / y))
	tmp = 0
	if (x - ((y * z) * t_1)) <= 1e+308:
		tmp = x - (z * (y * t_1))
	else:
		tmp = z * (t - x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(tanh(Float64(x / y)) - tanh(Float64(t / y)))
	tmp = 0.0
	if (Float64(x - Float64(Float64(y * z) * t_1)) <= 1e+308)
		tmp = Float64(x - Float64(z * Float64(y * t_1)));
	else
		tmp = Float64(z * Float64(t - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = tanh((x / y)) - tanh((t / y));
	tmp = 0.0;
	if ((x - ((y * z) * t_1)) <= 1e+308)
		tmp = x - (z * (y * t_1));
	else
		tmp = z * (t - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x - N[(N[(y * z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 1e+308], N[(x - N[(z * N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - x), $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))))) < 1e308

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   if 1e308 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 40.6%

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

      1. cancel-sign-sub 40.6%

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

      2. distribute-lft-neg-out 40.6%

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

      3. distribute-rgt-neg-in 40.6%

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

      4. neg-sub0 40.6%

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

      5. associate-+l- 40.6%

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

      6. neg-sub0 40.6%

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

      7. +-commutative 40.6%

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

      8. sub-neg 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in t around 0 37.6%

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

      1. associate-*r* 0.0%

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

      2. associate-/r* 0.0%

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

      3. rec-exp 0.0%

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

      4. div-sub 0.0%

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

      5. rec-exp 0.0%

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

      6. tanh-def-a 34.6%

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

   7. Simplified 34.6%

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

   8. Taylor expanded in z around inf 37.2%

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

      1. associate-/l* 37.2%

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

      2. rec-exp 37.2%

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

      3. rec-exp 37.2%

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

      4. tanh-def-a 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in y around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \leq 10^{+308}:\\ \;\;\;\;x - z \cdot \left(y \cdot \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq 1.15 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t\_1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (tanh (/ t y)))))
   (if (<= y 1.15e+41) (fma t_1 z x) (+ x (* z (- t_1 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * tanh((t / y));
	double tmp;
	if (y <= 1.15e+41) {
		tmp = fma(t_1, z, x);
	} else {
		tmp = x + (z * (t_1 - x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * tanh(Float64(t / y)))
	tmp = 0.0
	if (y <= 1.15e+41)
		tmp = fma(t_1, z, x);
	else
		tmp = Float64(x + Float64(z * Float64(t_1 - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.15e+41], N[(t$95$1 * z + x), $MachinePrecision], N[(x + N[(z * N[(t$95$1 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1499999999999999e41

   1. Initial program 95.2%

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

      1. *-commutative 95.2%

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

      2. associate-*l* 97.4%

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

   3. Simplified 97.4%

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

      1. *-un-lft-identity 97.4%

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

      2. *-commutative 97.4%

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

      3. +-commutative 97.4%

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

      4. *-commutative 97.4%

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

      5. fma-define 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. *-rgt-identity 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in x around 0 24.7%

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

       1. associate-/r* 24.7%

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

       2. rec-exp 24.7%

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

       3. div-sub 24.7%

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

       4. rec-exp 24.7%

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

       5. tanh-def-a 87.0%

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

   11. Simplified 87.0%

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

   if 1.1499999999999999e41 < y

   1. Initial program 91.3%

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

      1. *-commutative 91.3%

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

      2. associate-*l* 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. mul-1-neg 62.6%

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

      3. unsub-neg 62.6%

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

   7. Simplified 96.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \tanh \left(\frac{t}{y}\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq 1.66 \cdot 10^{+43}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot t\_1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))))
   (if (<= y 1.66e+43) (+ x (* (* y z) t_1)) (+ x (* z (- (* y t_1) x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double tmp;
	if (y <= 1.66e+43) {
		tmp = x + ((y * z) * t_1);
	} else {
		tmp = x + (z * ((y * t_1) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = tanh((t / y))
    if (y <= 1.66d+43) then
        tmp = x + ((y * z) * t_1)
    else
        tmp = x + (z * ((y * t_1) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.tanh((t / y));
	double tmp;
	if (y <= 1.66e+43) {
		tmp = x + ((y * z) * t_1);
	} else {
		tmp = x + (z * ((y * t_1) - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.tanh((t / y))
	tmp = 0
	if y <= 1.66e+43:
		tmp = x + ((y * z) * t_1)
	else:
		tmp = x + (z * ((y * t_1) - x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	tmp = 0.0
	if (y <= 1.66e+43)
		tmp = Float64(x + Float64(Float64(y * z) * t_1));
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y * t_1) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = tanh((t / y));
	tmp = 0.0;
	if (y <= 1.66e+43)
		tmp = x + ((y * z) * t_1);
	else
		tmp = x + (z * ((y * t_1) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.66e+43], N[(x + N[(N[(y * z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(N[(y * t$95$1), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6600000000000001e43

   1. Initial program 95.2%

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

      1. *-commutative 95.2%

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

      2. associate-*l* 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 24.7%

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

      1. associate-*r* 24.4%

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

      2. associate-/r* 24.4%

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

      3. div-sub 24.4%

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

      4. rec-exp 24.4%

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

      5. rec-exp 24.4%

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

      6. tanh-def-a 86.1%

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

      7. *-commutative 86.1%

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

   7. Simplified 86.1%

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

   if 1.6600000000000001e43 < y

   1. Initial program 91.3%

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

      1. *-commutative 91.3%

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

      2. associate-*l* 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. mul-1-neg 62.6%

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

      3. unsub-neg 62.6%

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

   7. Simplified 96.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.66 \cdot 10^{+43}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \tanh \left(\frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+172}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \tanh \left(\frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5.1e+172) (+ x (* (* y z) (tanh (/ t y)))) (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.1e+172) {
		tmp = x + ((y * z) * tanh((t / y)));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 5.1d+172) then
        tmp = x + ((y * z) * tanh((t / y)))
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.1e+172) {
		tmp = x + ((y * z) * Math.tanh((t / y)));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 5.1e+172:
		tmp = x + ((y * z) * math.tanh((t / y)))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.1e+172)
		tmp = Float64(x + Float64(Float64(y * z) * tanh(Float64(t / y))));
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 5.1e+172)
		tmp = x + ((y * z) * tanh((t / y)));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.1e+172], N[(x + N[(N[(y * z), $MachinePrecision] * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1e172

   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. *-commutative 95.3%

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

      2. associate-*l* 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 27.5%

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

      1. associate-*r* 27.3%

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

      2. associate-/r* 27.3%

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

      3. div-sub 27.3%

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

      4. rec-exp 27.3%

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

      5. rec-exp 27.3%

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

      6. tanh-def-a 85.8%

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

      7. *-commutative 85.8%

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

   7. Simplified 85.8%

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

   if 5.1e172 < y

   1. Initial program 85.9%

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

      1. cancel-sign-sub 85.9%

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

      2. distribute-lft-neg-out 85.9%

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

      3. distribute-rgt-neg-in 85.9%

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

      4. neg-sub0 85.9%

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

      5. associate-+l- 85.9%

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

      6. neg-sub0 85.9%

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

      7. +-commutative 85.9%

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

      8. sub-neg 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in y around inf 99.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+172}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \tanh \left(\frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+225}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.2e+38) x (if (<= y 1.85e+225) (- x (* x z)) (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.2e+38) {
		tmp = x;
	} else if (y <= 1.85e+225) {
		tmp = x - (x * z);
	} else {
		tmp = 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 <= 3.2d+38) then
        tmp = x
    else if (y <= 1.85d+225) then
        tmp = x - (x * z)
    else
        tmp = 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 <= 3.2e+38) {
		tmp = x;
	} else if (y <= 1.85e+225) {
		tmp = x - (x * z);
	} else {
		tmp = z * (t - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.2e+38:
		tmp = x
	elif y <= 1.85e+225:
		tmp = x - (x * z)
	else:
		tmp = z * (t - x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.2e+38)
		tmp = x;
	elseif (y <= 1.85e+225)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = Float64(z * Float64(t - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.2e+38)
		tmp = x;
	elseif (y <= 1.85e+225)
		tmp = x - (x * z);
	else
		tmp = z * (t - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.2e+38], x, If[LessEqual[y, 1.85e+225], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.19999999999999985e38

   1. Initial program 95.2%

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

      1. *-commutative 95.2%

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

      2. associate-*l* 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around inf 71.0%

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

   if 3.19999999999999985e38 < y < 1.84999999999999997e225

   1. Initial program 95.5%

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

      1. cancel-sign-sub 95.5%

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

      2. distribute-lft-neg-out 95.5%

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

      3. distribute-rgt-neg-in 95.5%

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

      4. neg-sub0 95.5%

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

      5. associate-+l- 95.5%

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

      6. neg-sub0 95.5%

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

      7. +-commutative 95.5%

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

      8. sub-neg 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in y around inf 80.3%

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

   6. Taylor expanded in x around inf 69.3%

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

   if 1.84999999999999997e225 < y

   1. Initial program 76.3%

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

      1. cancel-sign-sub 76.3%

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

      2. distribute-lft-neg-out 76.3%

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

      3. distribute-rgt-neg-in 76.3%

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

      4. neg-sub0 76.3%

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

      5. associate-+l- 76.3%

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

      6. neg-sub0 76.3%

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

      7. +-commutative 76.3%

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

      8. sub-neg 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in t around 0 76.3%

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

      1. associate-*r* 41.7%

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

      2. associate-/r* 41.7%

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

      3. rec-exp 41.7%

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

      4. div-sub 41.7%

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

      5. rec-exp 41.7%

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

      6. tanh-def-a 59.1%

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

   7. Simplified 59.1%

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

   8. Taylor expanded in z around inf 59.8%

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

      1. associate-/l* 59.8%

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

      2. rec-exp 59.8%

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

      3. rec-exp 59.8%

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

      4. tanh-def-a 75.6%

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

   10. Simplified 75.6%

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

   11. Taylor expanded in y around inf 75.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+225}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.5% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 32000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 32000.0) x (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 32000.0) {
		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 <= 32000.0d0) 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 <= 32000.0) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 32000.0:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 32000.0)
		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 <= 32000.0)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 32000.0], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 32000

   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. *-commutative 95.1%

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

      2. associate-*l* 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around inf 70.9%

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

   if 32000 < y

   1. Initial program 91.9%

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

      1. cancel-sign-sub 91.9%

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

      2. distribute-lft-neg-out 91.9%

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

      3. distribute-rgt-neg-in 91.9%

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

      4. neg-sub0 91.9%

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

      5. associate-+l- 91.9%

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

      6. neg-sub0 91.9%

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

      7. +-commutative 91.9%

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

      8. sub-neg 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around inf 87.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 32000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+169}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 4.6e+169) x (* z (- t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.6e+169) {
		tmp = x;
	} else {
		tmp = 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 <= 4.6d+169) then
        tmp = x
    else
        tmp = 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 <= 4.6e+169) {
		tmp = x;
	} else {
		tmp = z * (t - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.6e+169:
		tmp = x
	else:
		tmp = z * (t - x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.6e+169)
		tmp = x;
	else
		tmp = Float64(z * Float64(t - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.6e+169)
		tmp = x;
	else
		tmp = z * (t - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.6e+169], x, N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5999999999999999e169

   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. *-commutative 95.3%

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

      2. associate-*l* 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around inf 68.9%

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

   if 4.5999999999999999e169 < y

   1. Initial program 85.9%

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

      1. cancel-sign-sub 85.9%

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

      2. distribute-lft-neg-out 85.9%

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

      3. distribute-rgt-neg-in 85.9%

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

      4. neg-sub0 85.9%

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

      5. associate-+l- 85.9%

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

      6. neg-sub0 85.9%

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

      7. +-commutative 85.9%

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

      8. sub-neg 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in t around 0 60.1%

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

      1. associate-*r* 44.5%

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

      2. associate-/r* 44.5%

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

      3. rec-exp 44.5%

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

      4. div-sub 44.5%

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

      5. rec-exp 44.5%

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

      6. tanh-def-a 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in z around inf 38.7%

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

      1. associate-/l* 38.7%

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

      2. rec-exp 38.7%

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

      3. rec-exp 38.7%

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

      4. tanh-def-a 69.5%

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

   10. Simplified 69.5%

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

   11. Taylor expanded in y around inf 71.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+169}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.5% accurate, 23.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+254}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z 5.8e+254) x (* x (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.8e+254) {
		tmp = x;
	} else {
		tmp = x * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 5.8d+254) then
        tmp = x
    else
        tmp = x * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.8e+254) {
		tmp = x;
	} else {
		tmp = x * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 5.8e+254:
		tmp = x
	else:
		tmp = x * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5.8e+254)
		tmp = x;
	else
		tmp = Float64(x * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 5.8e+254)
		tmp = x;
	else
		tmp = x * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 5.8e+254], x, N[(x * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.7999999999999999e254

   1. Initial program 94.8%

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

      1. *-commutative 94.8%

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

      2. associate-*l* 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in x around inf 68.3%

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

   if 5.7999999999999999e254 < z

   1. Initial program 86.1%

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

      1. cancel-sign-sub 86.1%

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

      2. distribute-lft-neg-out 86.1%

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

      3. distribute-rgt-neg-in 86.1%

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

      4. neg-sub0 86.1%

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

      5. associate-+l- 86.1%

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

      6. neg-sub0 86.1%

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

      7. +-commutative 86.1%

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

      8. sub-neg 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in t around 0 23.1%

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

      1. associate-*r* 1.1%

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

      2. associate-/r* 1.1%

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

      3. rec-exp 1.1%

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

      4. div-sub 1.1%

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

      5. rec-exp 1.1%

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

      6. tanh-def-a 31.0%

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

   7. Simplified 31.0%

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

   8. Taylor expanded in z around inf 23.5%

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

      1. associate-/l* 23.5%

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

      2. rec-exp 23.5%

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

      3. rec-exp 23.5%

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

      4. tanh-def-a 57.2%

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

   10. Simplified 57.2%

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

   11. Taylor expanded in y around inf 66.3%

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

   12. Taylor expanded in t around 0 44.4%

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

       1. mul-1-neg 44.4%

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

       2. distribute-lft-neg-out 44.4%

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

       3. *-commutative 44.4%

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

   14. Simplified 44.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+254}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.0% 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 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. *-commutative 94.3%

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

   2. associate-*l* 97.7%

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

3. Simplified 97.7%

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

5. Taylor expanded in x around inf 65.0%

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

6. Final simplification 65.0%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 97.2% 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 2024048 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :alt
  (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))

  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))