SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.1% → 97.6%

Time: 13.1s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.1% 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: 97.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + 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. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. fma-define N/A

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

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot y\right), \color{blue}{z}, x\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\tanh \left(\frac{t}{y}\right), \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

   8. tanh-lowering-tanh.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\left(\frac{t}{y}\right)\right), \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

   10. tanh-lowering-tanh.f64 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{tanh.f64}\left(\left(\frac{x}{y}\right)\right)\right), y\right), z, x\right) \]

   11. /-lowering-/.f64 99.1%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(x, y\right)\right)\right), y\right), z, x\right) \]

4. Applied egg-rr 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{+94}:\\ \;\;\;\;x + \left(t\_1 - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(t\_1 - \frac{x}{y}\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))))
   (if (<= y 4.5e+94)
     (+ x (* (- t_1 (tanh (/ x y))) (* y z)))
     (fma (* y (- t_1 (/ x y))) z x))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double tmp;
	if (y <= 4.5e+94) {
		tmp = x + ((t_1 - tanh((x / y))) * (y * z));
	} else {
		tmp = fma((y * (t_1 - (x / y))), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	tmp = 0.0
	if (y <= 4.5e+94)
		tmp = Float64(x + Float64(Float64(t_1 - tanh(Float64(x / y))) * Float64(y * z)));
	else
		tmp = fma(Float64(y * Float64(t_1 - Float64(x / y))), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.5e+94], N[(x + N[(N[(t$95$1 - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(t$95$1 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.49999999999999972e94

   1. Initial program 97.7%

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

   if 4.49999999999999972e94 < y

   1. Initial program 88.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. fma-define N/A

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

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot y\right), \color{blue}{z}, x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\tanh \left(\frac{t}{y}\right), \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

      8. tanh-lowering-tanh.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\left(\frac{t}{y}\right)\right), \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

      10. tanh-lowering-tanh.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{tanh.f64}\left(\left(\frac{x}{y}\right)\right)\right), y\right), z, x\right) \]

      11. /-lowering-/.f64 97.2%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(x, y\right)\right)\right), y\right), z, x\right) \]

   4. Applied egg-rr 97.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \color{blue}{\left(\frac{x}{y}\right)}\right), y\right), z, x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 92.7%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{/.f64}\left(x, y\right)\right), y\right), z, x\right) \]

   7. Simplified 92.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+94}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 9.5e-139) x (fma (* y (- (tanh (/ t y)) (/ x y))) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e-139) {
		tmp = x;
	} else {
		tmp = fma((y * (tanh((t / y)) - (x / y))), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.5e-139)
		tmp = x;
	else
		tmp = fma(Float64(y * Float64(tanh(Float64(t / y)) - Float64(x / y))), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9.5e-139], x, N[(N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5000000000000006e-139

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 62.6%

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

   if 9.5000000000000006e-139 < y

   1. Initial program 94.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. fma-define N/A

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

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot y\right), \color{blue}{z}, x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\tanh \left(\frac{t}{y}\right), \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

      8. tanh-lowering-tanh.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\left(\frac{t}{y}\right)\right), \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \tanh \left(\frac{x}{y}\right)\right), y\right), z, x\right) \]

      10. tanh-lowering-tanh.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{tanh.f64}\left(\left(\frac{x}{y}\right)\right)\right), y\right), z, x\right) \]

      11. /-lowering-/.f64 98.6%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(x, y\right)\right)\right), y\right), z, x\right) \]

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \color{blue}{\left(\frac{x}{y}\right)}\right), y\right), z, x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 87.3%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{/.f64}\left(x, y\right)\right), y\right), z, x\right) \]

   7. Simplified 87.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * ((tanh((t / y)) - tanh((x / y))) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(y * N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \cdot \color{blue}{y}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right), \color{blue}{y}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right), y\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\tanh \left(\frac{t}{y}\right), \tanh \left(\frac{x}{y}\right)\right)\right), y\right)\right) \]

   6. tanh-lowering-tanh.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\left(\frac{t}{y}\right)\right), \tanh \left(\frac{x}{y}\right)\right)\right), y\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \tanh \left(\frac{x}{y}\right)\right)\right), y\right)\right) \]

   8. tanh-lowering-tanh.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{tanh.f64}\left(\left(\frac{x}{y}\right)\right)\right)\right), y\right)\right) \]

   9. /-lowering-/.f64 98.9%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(x, y\right)\right)\right)\right), y\right)\right) \]

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 5: 71.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+128}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right) - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.15e-145)
   x
   (if (<= y 4.8e+128)
     (+ x (- (* (tanh (/ t y)) (* y z)) (* x z)))
     (+ x (* z (- t x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.15e-145) {
		tmp = x;
	} else if (y <= 4.8e+128) {
		tmp = x + ((tanh((t / y)) * (y * z)) - (x * z));
	} 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 <= 1.15d-145) then
        tmp = x
    else if (y <= 4.8d+128) then
        tmp = x + ((tanh((t / y)) * (y * z)) - (x * z))
    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 <= 1.15e-145) {
		tmp = x;
	} else if (y <= 4.8e+128) {
		tmp = x + ((Math.tanh((t / y)) * (y * z)) - (x * z));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.15e-145:
		tmp = x
	elif y <= 4.8e+128:
		tmp = x + ((math.tanh((t / y)) * (y * z)) - (x * z))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.15e-145)
		tmp = x;
	elseif (y <= 4.8e+128)
		tmp = Float64(x + Float64(Float64(tanh(Float64(t / y)) * Float64(y * z)) - Float64(x * z)));
	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 <= 1.15e-145)
		tmp = x;
	elseif (y <= 4.8e+128)
		tmp = x + ((tanh((t / y)) * (y * z)) - (x * z));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.15e-145], x, If[LessEqual[y, 4.8e+128], N[(x + N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15000000000000004e-145

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 62.6%

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

   if 1.15000000000000004e-145 < y < 4.8000000000000004e128

   1. Initial program 99.8%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) + \color{blue}{\left(\mathsf{neg}\left(\tanh \left(\frac{x}{y}\right)\right)\right)}\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(y \cdot z\right) \cdot \tanh \left(\frac{t}{y}\right) + \color{blue}{\left(y \cdot z\right) \cdot \left(\mathsf{neg}\left(\tanh \left(\frac{x}{y}\right)\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\left(\left(y \cdot z\right) \cdot \tanh \left(\frac{t}{y}\right)\right), \color{blue}{\left(\left(y \cdot z\right) \cdot \left(\mathsf{neg}\left(\tanh \left(\frac{x}{y}\right)\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(y \cdot z\right), \tanh \left(\frac{t}{y}\right)\right), \left(\color{blue}{\left(y \cdot z\right)} \cdot \left(\mathsf{neg}\left(\tanh \left(\frac{x}{y}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \tanh \left(\frac{t}{y}\right)\right), \left(\left(\color{blue}{y} \cdot z\right) \cdot \left(\mathsf{neg}\left(\tanh \left(\frac{x}{y}\right)\right)\right)\right)\right)\right) \]

      6. tanh-lowering-tanh.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\left(\frac{t}{y}\right)\right)\right), \left(\left(y \cdot \color{blue}{z}\right) \cdot \left(\mathsf{neg}\left(\tanh \left(\frac{x}{y}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \left(\left(y \cdot z\right) \cdot \left(\mathsf{neg}\left(\tanh \left(\frac{x}{y}\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(\tanh \left(\frac{x}{y}\right)\right)\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{\tanh \left(\frac{x}{y}\right)}\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(0 - \color{blue}{\tanh \left(\frac{x}{y}\right)}\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{\_.f64}\left(0, \color{blue}{\tanh \left(\frac{x}{y}\right)}\right)\right)\right)\right) \]

      12. tanh-lowering-tanh.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{\_.f64}\left(0, \mathsf{tanh.f64}\left(\left(\frac{x}{y}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 97.9%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{\_.f64}\left(0, \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(x, y\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \color{blue}{\left(-1 \cdot \left(x \cdot z\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \left(\mathsf{neg}\left(x \cdot z\right)\right)\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \left(x \cdot \left(-1 \cdot \color{blue}{z}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot z\right)}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(x, \left(0 - \color{blue}{z}\right)\right)\right)\right) \]

      7. --lowering--.f64 86.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{z}\right)\right)\right)\right) \]

   7. Simplified 86.0%

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

   if 4.8000000000000004e128 < y

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(z \cdot \left(t - x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(t - x\right)}\right)\right) \]

      3. --lowering--.f64 94.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 94.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+128}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right) - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+180}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 7.2e-140)
   x
   (if (<= y 4.8e+180)
     (+ x (* (* y z) (- (tanh (/ t y)) (/ x y))))
     (+ x (* z (- t x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.2e-140) {
		tmp = x;
	} else if (y <= 4.8e+180) {
		tmp = x + ((y * z) * (tanh((t / y)) - (x / 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 <= 7.2d-140) then
        tmp = x
    else if (y <= 4.8d+180) then
        tmp = x + ((y * z) * (tanh((t / y)) - (x / 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 <= 7.2e-140) {
		tmp = x;
	} else if (y <= 4.8e+180) {
		tmp = x + ((y * z) * (Math.tanh((t / y)) - (x / y)));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 7.2e-140:
		tmp = x
	elif y <= 4.8e+180:
		tmp = x + ((y * z) * (math.tanh((t / y)) - (x / y)))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7.2e-140)
		tmp = x;
	elseif (y <= 4.8e+180)
		tmp = Float64(x + Float64(Float64(y * z) * Float64(tanh(Float64(t / y)) - Float64(x / 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 <= 7.2e-140)
		tmp = x;
	elseif (y <= 4.8e+180)
		tmp = x + ((y * z) * (tanh((t / y)) - (x / y)));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 7.2e-140], x, If[LessEqual[y, 4.8e+180], N[(x + N[(N[(y * z), $MachinePrecision] * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.2000000000000001e-140

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 62.6%

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

   if 7.2000000000000001e-140 < y < 4.7999999999999997e180

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \color{blue}{\left(\frac{x}{y}\right)}\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 83.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{\_.f64}\left(\mathsf{tanh.f64}\left(\mathsf{/.f64}\left(t, y\right)\right), \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 83.4%

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

   if 4.7999999999999997e180 < y

   1. Initial program 87.0%

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

   3. Taylor expanded in y around inf

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(z \cdot \left(t - x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(t - x\right)}\right)\right) \]

      3. --lowering--.f64 95.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 95.5%

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

Alternative 7: 69.0% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.8e-72) x (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.8e-72) {
		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 <= 3.8d-72) 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 <= 3.8e-72) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.8e-72:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.8e-72)
		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 <= 3.8e-72)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.8e-72], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.80000000000000002e-72

   1. Initial program 97.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 62.3%

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

   if 3.80000000000000002e-72 < y

   1. Initial program 93.6%

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

   3. Taylor expanded in y around inf

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(z \cdot \left(t - x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(t - x\right)}\right)\right) \]

      3. --lowering--.f64 76.7%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(t, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 76.7%

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

Alternative 8: 65.2% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.05e-72) x (+ x (* t z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.05e-72) {
		tmp = x;
	} else {
		tmp = x + (t * 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 <= 2.05d-72) then
        tmp = x
    else
        tmp = x + (t * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.05e-72) {
		tmp = x;
	} else {
		tmp = x + (t * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.05e-72:
		tmp = x
	else:
		tmp = x + (t * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.05e-72)
		tmp = x;
	else
		tmp = Float64(x + Float64(t * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.05e-72)
		tmp = x;
	else
		tmp = x + (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.05e-72], x, N[(x + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.05000000000000002e-72

   1. Initial program 97.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 62.3%

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

   if 2.05000000000000002e-72 < y

   1. Initial program 93.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \color{blue}{\left(\frac{t - x}{y}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(\left(t - x\right), \color{blue}{y}\right)\right)\right) \]

      2. --lowering--.f64 67.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, x\right), y\right)\right)\right) \]

   5. Simplified 67.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(t \cdot z\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(z \cdot \color{blue}{t}\right)\right) \]

      2. *-lowering-*.f64 66.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{t}\right)\right) \]

   8. Simplified 66.5%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.1% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-45}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.4e-45) x (* x (- 1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.4e-45) {
		tmp = 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 <= 2.4d-45) then
        tmp = 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 <= 2.4e-45) {
		tmp = x;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.4e-45:
		tmp = x
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.4e-45)
		tmp = 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 <= 2.4e-45)
		tmp = x;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.4e-45], x, N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3999999999999999e-45

   1. Initial program 97.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 62.0%

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

   if 2.3999999999999999e-45 < y

   1. Initial program 93.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \color{blue}{\left(\frac{t - x}{y}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(\left(t - x\right), \color{blue}{y}\right)\right)\right) \]

      2. --lowering--.f64 68.7%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, x\right), y\right)\right)\right) \]

   5. Simplified 68.7%

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

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot z\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{z}\right)\right) \]

      4. --lowering--.f64 54.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{z}\right)\right) \]

   8. Simplified 54.8%

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

Alternative 10: 60.6% 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

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 57.1%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 96.8% 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 2024161 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y)))))))

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