SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.3% → 97.8%

Time: 12.6s

Alternatives: 13

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: 93.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. +-commutative 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*l* 98.8%

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

   4. fma-def 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{x}{y}\right)\\ t_2 := x + \left(\tanh \left(\frac{t}{y}\right) - t_1\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t_2 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - y \cdot t_1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ x y))) (t_2 (+ x (* (- (tanh (/ t y)) t_1) (* z y)))))
   (if (<= t_2 2e+299) t_2 (fma z (- t (* y t_1)) x))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((x / y));
	double t_2 = x + ((tanh((t / y)) - t_1) * (z * y));
	double tmp;
	if (t_2 <= 2e+299) {
		tmp = t_2;
	} else {
		tmp = fma(z, (t - (y * t_1)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = tanh(Float64(x / y))
	t_2 = Float64(x + Float64(Float64(tanh(Float64(t / y)) - t_1) * Float64(z * y)))
	tmp = 0.0
	if (t_2 <= 2e+299)
		tmp = t_2;
	else
		tmp = fma(z, Float64(t - Float64(y * t_1)), 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[(x + N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e+299], t$95$2, N[(z * N[(t - N[(y * t$95$1), $MachinePrecision]), $MachinePrecision] + x), $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))))) < 2.0000000000000001e299

   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. Add Preprocessing

   if 2.0000000000000001e299 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 55.6%

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

      1. +-commutative 55.6%

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

      2. *-commutative 55.6%

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

      3. associate-*l* 84.9%

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

      4. fma-def 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in t around 0 77.5%

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

   6. Taylor expanded in t around -inf 61.6%

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

      1. mul-1-neg 61.6%

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

      2. unsub-neg 61.6%

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

      3. associate-/r* 61.6%

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

      4. rec-exp 61.6%

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

      5. div-sub 61.6%

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

   8. Simplified 92.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{+299}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - y \cdot \tanh \left(\frac{x}{y}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5400000000 \lor \neg \left(t \leq 1.85 \cdot 10^{+18}\right):\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - y \cdot \tanh \left(\frac{x}{y}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -5400000000.0) (not (<= t 1.85e+18)))
   (+ x (* z (* y (tanh (/ t y)))))
   (fma z (- t (* y (tanh (/ x y)))) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5400000000.0) || !(t <= 1.85e+18)) {
		tmp = x + (z * (y * tanh((t / y))));
	} else {
		tmp = fma(z, (t - (y * tanh((x / y)))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -5400000000.0) || !(t <= 1.85e+18))
		tmp = Float64(x + Float64(z * Float64(y * tanh(Float64(t / y)))));
	else
		tmp = fma(z, Float64(t - Float64(y * tanh(Float64(x / y)))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -5400000000.0], N[Not[LessEqual[t, 1.85e+18]], $MachinePrecision]], N[(x + N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - N[(y * N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.4e9 or 1.85e18 < t

   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. Add Preprocessing

   3. Taylor expanded in x around 0 10.1%

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

      1. associate-/r* 10.1%

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

      2. div-sub 10.1%

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

      3. rec-exp 10.1%

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

      4. rec-exp 10.1%

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

      5. tanh-def-a 85.2%

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

   5. Simplified 85.2%

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

      1. add-log-exp 55.7%

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

      2. associate-*r* 55.7%

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

      3. exp-prod 52.5%

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

      4. *-commutative 52.5%

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

      5. exp-prod 43.5%

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

   7. Applied egg-rr 43.5%

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

      1. log-pow 43.5%

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

      2. log-pow 45.7%

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

      3. rem-log-exp 83.8%

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

   9. Simplified 83.8%

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

   10. Taylor expanded in t around inf 10.1%

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

       1. associate-*r* 10.0%

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

       2. associate-*r/ 10.0%

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

       3. rec-exp 10.0%

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

       4. rec-exp 10.0%

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

       5. tanh-def-a 83.8%

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

       6. *-commutative 83.8%

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

       7. associate-*l* 86.0%

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

   12. Simplified 86.0%

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

   if -5.4e9 < t < 1.85e18

   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. +-commutative 95.5%

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

      2. *-commutative 95.5%

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

      3. associate-*l* 97.8%

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

      4. fma-def 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in t around 0 89.6%

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

   6. Taylor expanded in t around -inf 31.2%

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

      1. mul-1-neg 31.2%

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

      2. unsub-neg 31.2%

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

      3. associate-/r* 31.2%

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

      4. rec-exp 31.2%

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

      5. div-sub 31.2%

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

   8. Simplified 91.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5400000000 \lor \neg \left(t \leq 1.85 \cdot 10^{+18}\right):\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - y \cdot \tanh \left(\frac{x}{y}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ t_2 := x + z \cdot \left(y \cdot t_1\right)\\ \mathbf{if}\;y \leq 2650000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.82 \cdot 10^{+170}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \left(t_1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+193}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))) (t_2 (+ x (* z (* y t_1)))))
   (if (<= y 2650000000.0)
     t_2
     (if (<= y 1.82e+170)
       (+ x (* (* z y) (- t_1 (/ x y))))
       (if (<= y 7.5e+193) t_2 (+ x (* z (- t x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double t_2 = x + (z * (y * t_1));
	double tmp;
	if (y <= 2650000000.0) {
		tmp = t_2;
	} else if (y <= 1.82e+170) {
		tmp = x + ((z * y) * (t_1 - (x / y)));
	} else if (y <= 7.5e+193) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = tanh((t / y))
    t_2 = x + (z * (y * t_1))
    if (y <= 2650000000.0d0) then
        tmp = t_2
    else if (y <= 1.82d+170) then
        tmp = x + ((z * y) * (t_1 - (x / y)))
    else if (y <= 7.5d+193) then
        tmp = t_2
    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 = Math.tanh((t / y));
	double t_2 = x + (z * (y * t_1));
	double tmp;
	if (y <= 2650000000.0) {
		tmp = t_2;
	} else if (y <= 1.82e+170) {
		tmp = x + ((z * y) * (t_1 - (x / y)));
	} else if (y <= 7.5e+193) {
		tmp = t_2;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.tanh((t / y))
	t_2 = x + (z * (y * t_1))
	tmp = 0
	if y <= 2650000000.0:
		tmp = t_2
	elif y <= 1.82e+170:
		tmp = x + ((z * y) * (t_1 - (x / y)))
	elif y <= 7.5e+193:
		tmp = t_2
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	t_2 = Float64(x + Float64(z * Float64(y * t_1)))
	tmp = 0.0
	if (y <= 2650000000.0)
		tmp = t_2;
	elseif (y <= 1.82e+170)
		tmp = Float64(x + Float64(Float64(z * y) * Float64(t_1 - Float64(x / y))));
	elseif (y <= 7.5e+193)
		tmp = t_2;
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = tanh((t / y));
	t_2 = x + (z * (y * t_1));
	tmp = 0.0;
	if (y <= 2650000000.0)
		tmp = t_2;
	elseif (y <= 1.82e+170)
		tmp = x + ((z * y) * (t_1 - (x / y)));
	elseif (y <= 7.5e+193)
		tmp = t_2;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(z * N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2650000000.0], t$95$2, If[LessEqual[y, 1.82e+170], N[(x + N[(N[(z * y), $MachinePrecision] * N[(t$95$1 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+193], t$95$2, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.65e9 or 1.82e170 < y < 7.5000000000000008e193

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0 22.1%

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

      1. associate-/r* 22.1%

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

      2. div-sub 22.1%

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

      3. rec-exp 22.1%

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

      4. rec-exp 22.1%

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

      5. tanh-def-a 82.6%

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

   5. Simplified 82.6%

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

      1. add-log-exp 60.7%

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

      2. associate-*r* 60.7%

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

      3. exp-prod 57.5%

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

      4. *-commutative 57.5%

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

      5. exp-prod 42.7%

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

   7. Applied egg-rr 42.7%

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

      1. log-pow 42.6%

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

      2. log-pow 43.6%

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

      3. rem-log-exp 82.6%

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

   9. Simplified 82.6%

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

   10. Taylor expanded in t around inf 22.1%

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

       1. associate-*r* 21.9%

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

       2. associate-*r/ 21.9%

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

       3. rec-exp 22.0%

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

       4. rec-exp 22.0%

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

       5. tanh-def-a 82.6%

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

       6. *-commutative 82.6%

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

       7. associate-*l* 84.1%

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

   12. Simplified 84.1%

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

   if 2.65e9 < y < 1.82e170

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0 83.1%

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

   if 7.5000000000000008e193 < y

   1. Initial program 88.1%

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

   3. Taylor expanded in y around inf 93.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2650000000:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.82 \cdot 10^{+170}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+193}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -15600000000000 \lor \neg \left(t \leq 2.8 \cdot 10^{-32}\right):\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -15600000000000.0) (not (<= t 2.8e-32)))
   (+ x (* z (* y (tanh (/ t y)))))
   (+ x (* (* z y) (- (/ t y) (tanh (/ x y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -15600000000000.0) || !(t <= 2.8e-32)) {
		tmp = x + (z * (y * tanh((t / y))));
	} else {
		tmp = x + ((z * y) * ((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 <= (-15600000000000.0d0)) .or. (.not. (t <= 2.8d-32))) then
        tmp = x + (z * (y * tanh((t / y))))
    else
        tmp = x + ((z * y) * ((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 <= -15600000000000.0) || !(t <= 2.8e-32)) {
		tmp = x + (z * (y * Math.tanh((t / y))));
	} else {
		tmp = x + ((z * y) * ((t / y) - Math.tanh((x / y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -15600000000000.0) or not (t <= 2.8e-32):
		tmp = x + (z * (y * math.tanh((t / y))))
	else:
		tmp = x + ((z * y) * ((t / y) - math.tanh((x / y))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -15600000000000.0) || !(t <= 2.8e-32))
		tmp = Float64(x + Float64(z * Float64(y * tanh(Float64(t / y)))));
	else
		tmp = Float64(x + Float64(Float64(z * y) * 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 <= -15600000000000.0) || ~((t <= 2.8e-32)))
		tmp = x + (z * (y * tanh((t / y))));
	else
		tmp = x + ((z * y) * ((t / y) - tanh((x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -15600000000000.0], N[Not[LessEqual[t, 2.8e-32]], $MachinePrecision]], N[(x + N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z * y), $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 < -1.56e13 or 2.7999999999999999e-32 < t

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0 10.4%

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

      1. associate-/r* 10.4%

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

      2. div-sub 10.4%

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

      3. rec-exp 10.4%

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

      4. rec-exp 10.4%

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

      5. tanh-def-a 83.1%

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

   5. Simplified 83.1%

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

      1. add-log-exp 53.9%

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

      2. associate-*r* 53.9%

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

      3. exp-prod 51.1%

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

      4. *-commutative 51.1%

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

      5. exp-prod 41.9%

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

   7. Applied egg-rr 41.9%

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

      1. log-pow 41.9%

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

      2. log-pow 43.8%

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

      3. rem-log-exp 82.5%

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

   9. Simplified 82.5%

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

   10. Taylor expanded in t around inf 10.4%

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

       1. associate-*r* 10.2%

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

       2. associate-*r/ 10.2%

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

       3. rec-exp 10.2%

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

       4. rec-exp 10.2%

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

       5. tanh-def-a 82.5%

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

       6. *-commutative 82.5%

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

       7. associate-*l* 84.5%

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

   12. Simplified 84.5%

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

   if -1.56e13 < t < 2.7999999999999999e-32

   1. Initial program 96.5%

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

   3. Taylor expanded in t around 0 91.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -15600000000000 \lor \neg \left(t \leq 2.8 \cdot 10^{-32}\right):\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+114}:\\ \;\;\;\;x + y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.1e+114) (+ x (* y (* z (tanh (/ t y))))) (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.1e+114) {
		tmp = x + (y * (z * tanh((t / y))));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.1d+114) then
        tmp = x + (y * (z * tanh((t / y))))
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.1e+114) {
		tmp = x + (y * (z * Math.tanh((t / y))));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.1e+114:
		tmp = x + (y * (z * math.tanh((t / y))))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.1e+114)
		tmp = Float64(x + Float64(y * Float64(z * tanh(Float64(t / y)))));
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.1e+114)
		tmp = x + (y * (z * tanh((t / y))));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.1e+114], N[(x + N[(y * N[(z * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e114

   1. Initial program 98.2%

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

   3. Taylor expanded in x around 0 23.8%

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

      1. associate-/r* 23.8%

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

      2. div-sub 23.8%

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

      3. rec-exp 23.8%

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

      4. rec-exp 23.8%

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

      5. tanh-def-a 82.1%

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

   5. Simplified 82.1%

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

   if 1.1e114 < y

   1. Initial program 85.5%

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

   3. Taylor expanded in y around inf 93.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+114}:\\ \;\;\;\;x + y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+115}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.7e+115) (+ 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 <= 2.7e+115) {
		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 <= 2.7d+115) 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 <= 2.7e+115) {
		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 <= 2.7e+115:
		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 <= 2.7e+115)
		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 <= 2.7e+115)
		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, 2.7e+115], 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 < 2.70000000000000004e115

   1. Initial program 98.2%

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

   3. Taylor expanded in x around 0 23.8%

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

      1. associate-/r* 23.8%

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

      2. div-sub 23.8%

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

      3. rec-exp 23.8%

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

      4. rec-exp 23.8%

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

      5. tanh-def-a 82.1%

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

   5. Simplified 82.1%

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

      1. add-log-exp 59.4%

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

      2. associate-*r* 59.4%

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

      3. exp-prod 56.1%

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

      4. *-commutative 56.1%

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

      5. exp-prod 43.6%

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

   7. Applied egg-rr 43.6%

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

      1. log-pow 43.5%

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

      2. log-pow 44.3%

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

      3. rem-log-exp 82.0%

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

   9. Simplified 82.0%

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

   10. Taylor expanded in t around inf 23.8%

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

       1. associate-*r* 23.7%

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

       2. associate-*r/ 23.7%

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

       3. rec-exp 23.7%

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

       4. rec-exp 23.7%

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

       5. tanh-def-a 82.0%

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

       6. *-commutative 82.0%

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

       7. associate-*l* 82.5%

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

   12. Simplified 82.5%

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

   if 2.70000000000000004e115 < y

   1. Initial program 85.5%

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

   3. Taylor expanded in y around inf 93.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+115}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.3% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2650000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+135} \lor \neg \left(y \leq 1.1 \cdot 10^{+182}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2650000000.0)
   x
   (if (or (<= y 1.42e+135) (not (<= y 1.1e+182))) (* x (- 1.0 z)) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2650000000.0) {
		tmp = x;
	} else if ((y <= 1.42e+135) || !(y <= 1.1e+182)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = 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 <= 2650000000.0d0) then
        tmp = x
    else if ((y <= 1.42d+135) .or. (.not. (y <= 1.1d+182))) then
        tmp = x * (1.0d0 - z)
    else
        tmp = 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 <= 2650000000.0) {
		tmp = x;
	} else if ((y <= 1.42e+135) || !(y <= 1.1e+182)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2650000000.0:
		tmp = x
	elif (y <= 1.42e+135) or not (y <= 1.1e+182):
		tmp = x * (1.0 - z)
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2650000000.0)
		tmp = x;
	elseif ((y <= 1.42e+135) || !(y <= 1.1e+182))
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2650000000.0)
		tmp = x;
	elseif ((y <= 1.42e+135) || ~((y <= 1.1e+182)))
		tmp = x * (1.0 - z);
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2650000000.0], x, If[Or[LessEqual[y, 1.42e+135], N[Not[LessEqual[y, 1.1e+182]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.65e9

   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. Add Preprocessing

   3. Taylor expanded in x around inf 67.7%

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

   if 2.65e9 < y < 1.41999999999999998e135 or 1.09999999999999998e182 < y

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf 75.1%

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

   4. Taylor expanded in x around inf 58.9%

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

      1. mul-1-neg 58.9%

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

      2. unsub-neg 58.9%

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

   6. Simplified 58.9%

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

   if 1.41999999999999998e135 < y < 1.09999999999999998e182

   1. Initial program 86.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   4. Taylor expanded in x around 0 72.3%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2650000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+135} \lor \neg \left(y \leq 1.1 \cdot 10^{+182}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.1% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+220}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.2e+105) x (if (<= y 7.2e+220) (* z (- t x)) (* x (- 1.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.2e+105) {
		tmp = x;
	} else if (y <= 7.2e+220) {
		tmp = z * (t - x);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 3.2d+105) then
        tmp = x
    else if (y <= 7.2d+220) then
        tmp = z * (t - x)
    else
        tmp = x * (1.0d0 - z)
    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+105) {
		tmp = x;
	} else if (y <= 7.2e+220) {
		tmp = z * (t - x);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.2e+105:
		tmp = x
	elif y <= 7.2e+220:
		tmp = z * (t - x)
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.2e+105)
		tmp = x;
	elseif (y <= 7.2e+220)
		tmp = Float64(z * Float64(t - x));
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.2e+105)
		tmp = x;
	elseif (y <= 7.2e+220)
		tmp = z * (t - x);
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.2e+105], x, If[LessEqual[y, 7.2e+220], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.2e105

   1. Initial program 98.2%

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

   3. Taylor expanded in x around inf 64.6%

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

   if 3.2e105 < y < 7.20000000000000038e220

   1. Initial program 86.5%

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

   3. Taylor expanded in y around inf 91.1%

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

   4. Taylor expanded in z around inf 72.6%

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

   if 7.20000000000000038e220 < y

   1. Initial program 85.8%

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

   3. Taylor expanded in y around inf 91.6%

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

   4. Taylor expanded in x around inf 72.5%

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

      1. mul-1-neg 72.5%

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

      2. unsub-neg 72.5%

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

   6. Simplified 72.5%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+220}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.48 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.48e-29) x (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.48e-29) {
		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 <= 1.48d-29) 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 <= 1.48e-29) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.48e-29:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.48e-29)
		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 <= 1.48e-29)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.48e-29], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4800000000000001e-29

   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. Add Preprocessing

   3. Taylor expanded in x around inf 67.9%

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

   if 1.4800000000000001e-29 < y

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf 75.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.48 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.7% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.48 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 1.48e-29) x (+ x (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.48e-29) {
		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 <= 1.48d-29) 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 <= 1.48e-29) {
		tmp = x;
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.48e-29:
		tmp = x
	else:
		tmp = x + (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.48e-29)
		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 <= 1.48e-29)
		tmp = x;
	else
		tmp = x + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.48e-29], x, N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4800000000000001e-29

   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. Add Preprocessing

   3. Taylor expanded in x around inf 67.9%

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

   if 1.4800000000000001e-29 < y

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. *-commutative 91.0%

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

      3. associate-*l* 97.4%

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

      4. fma-def 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 32.4%

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

      1. associate-/r* 32.4%

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

      2. div-sub 32.4%

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

      3. rec-exp 32.4%

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

      4. rec-exp 32.4%

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

      5. tanh-def-a 73.4%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in y around inf 64.2%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.48 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 26.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{+230}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z 5.4e+230) x (* z t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.4e+230) {
		tmp = x;
	} else {
		tmp = 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 (z <= 5.4d+230) then
        tmp = x
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5.4e+230) {
		tmp = x;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 5.4e+230:
		tmp = x
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5.4e+230)
		tmp = x;
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 5.4e+230)
		tmp = x;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 5.4e+230], x, N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.40000000000000006e230

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf 63.8%

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

   if 5.40000000000000006e230 < z

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf 83.2%

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

   4. Taylor expanded in x around 0 65.8%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{+230}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.7% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in x around inf 60.1%

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

4. Final simplification 60.1%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024020 
(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))))))