SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.6% → 99.0%

Time: 11.4s

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: 93.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(t - x\right)\\ t_2 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- t x)))
        (t_2 (+ x (* (- (tanh (/ t y)) (tanh (/ x y))) (* y z)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+307) t_2 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (t - x);
	double t_2 = x + ((tanh((t / y)) - tanh((x / y))) * (y * z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+307) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (t - x);
	double t_2 = x + ((Math.tanh((t / y)) - Math.tanh((x / y))) * (y * z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 5e+307) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (t - x)
	t_2 = x + ((math.tanh((t / y)) - math.tanh((x / y))) * (y * z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 5e+307:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(t - x))
	t_2 = Float64(x + Float64(Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))) * Float64(y * z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+307)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (t - x);
	t_2 = x + ((tanh((t / y)) - tanh((x / y))) * (y * z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 5e+307)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+307], t$95$2, t$95$1]]]]

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))))) < -inf.0 or 5e307 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 56.4%

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

   3. Taylor expanded in 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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf

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

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

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

      2. --lowering--.f64 100.0%

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

   8. Simplified 100.0%

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 5e307

   1. Initial program 98.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{elif}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% 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 94.9%

   \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
2. Add Preprocessing
3. 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 97.9%

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

4. Applied egg-rr 97.9%

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

5. Final simplification 97.9%

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

Alternative 3: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5.2e-52) 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 <= 5.2e-52) {
		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 <= 5.2e-52)
		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, 5.2e-52], 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 < 5.1999999999999997e-52

   1. Initial program 97.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 69.5%

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

   if 5.1999999999999997e-52 < y

   1. Initial program 87.9%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   2. Add Preprocessing
   3. 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 95.5%

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

   4. Applied egg-rr 95.5%

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

   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 86.7%

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

   7. Simplified 86.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.0× speedup

Math

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

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(Float64(y * 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[(N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f64 97.9%

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

4. Applied egg-rr 97.9%

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

5. Final simplification 97.9%

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

Alternative 5: 68.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+66}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.45e-115)
   x
   (if (<= y 9.2e+66)
     (+ x (* (* y z) (- (/ t y) (tanh (/ x y)))))
     (+ x (* z (- t x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.45e-115) {
		tmp = x;
	} else if (y <= 9.2e+66) {
		tmp = x + ((y * z) * ((t / y) - tanh((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 <= 3.45d-115) then
        tmp = x
    else if (y <= 9.2d+66) then
        tmp = x + ((y * z) * ((t / y) - tanh((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 <= 3.45e-115) {
		tmp = x;
	} else if (y <= 9.2e+66) {
		tmp = x + ((y * z) * ((t / y) - Math.tanh((x / y))));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.45e-115:
		tmp = x
	elif y <= 9.2e+66:
		tmp = x + ((y * z) * ((t / y) - math.tanh((x / y))))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.45e-115)
		tmp = x;
	elseif (y <= 9.2e+66)
		tmp = Float64(x + Float64(Float64(y * z) * Float64(Float64(t / y) - tanh(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 <= 3.45e-115)
		tmp = x;
	elseif (y <= 9.2e+66)
		tmp = x + ((y * z) * ((t / y) - tanh((x / y))));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.45e-115], x, If[LessEqual[y, 9.2e+66], N[(x + N[(N[(y * z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.44999999999999999e-115

   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

   3. Taylor expanded in x around inf

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

   5. Simplified 69.6%

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

   if 3.44999999999999999e-115 < y < 9.2e66

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 69.9%

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

   5. Simplified 69.9%

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

   if 9.2e66 < y

   1. Initial program 80.7%

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

   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 82.0%

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

   5. Simplified 82.0%

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

Alternative 6: 71.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5e-52) x (+ x (* z (* y (- (tanh (/ t y)) (/ x y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5e-52) {
		tmp = x;
	} else {
		tmp = x + (z * (y * (tanh((t / y)) - (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 (y <= 5d-52) then
        tmp = x
    else
        tmp = x + (z * (y * (tanh((t / y)) - (x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5e-52) {
		tmp = x;
	} else {
		tmp = x + (z * (y * (Math.tanh((t / y)) - (x / y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 5e-52:
		tmp = x
	else:
		tmp = x + (z * (y * (math.tanh((t / y)) - (x / y))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5e-52)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * Float64(y * Float64(tanh(Float64(t / y)) - Float64(x / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 5e-52)
		tmp = x;
	else
		tmp = x + (z * (y * (tanh((t / y)) - (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5e-52], x, N[(x + N[(z * N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e-52

   1. Initial program 97.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 69.5%

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

   if 5e-52 < y

   1. Initial program 87.9%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   2. Add Preprocessing
   3. 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 95.5%

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

   4. Applied egg-rr 95.5%

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

   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 86.7%

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

   7. Simplified 86.7%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 79.1%

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

   9. Applied egg-rr 79.1%

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

       1. associate-*r* N/A

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

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

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

       3. *-commutative N/A

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

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

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

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

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

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

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

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

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

       8. /-lowering-/.f64 86.7%

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

   11. Applied egg-rr 86.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.12e+22) x (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.12e+22) {
		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 <= 1.12d+22) 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 <= 1.12e+22) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.12e+22:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.12e+22)
		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 <= 1.12e+22)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.12e+22], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.12e22

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 68.6%

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

   if 1.12e22 < y

   1. Initial program 84.4%

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

   3. Taylor expanded in 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 75.9%

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

   5. Simplified 75.9%

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

Alternative 8: 65.2% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 16000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 16000.0) x (+ x (* t z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 16000.0) {
		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 <= 16000.0d0) 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 <= 16000.0) {
		tmp = x;
	} else {
		tmp = x + (t * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 16000.0:
		tmp = x
	else:
		tmp = x + (t * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 16000.0)
		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 <= 16000.0)
		tmp = x;
	else
		tmp = x + (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 16000.0], x, N[(x + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 16000

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 68.2%

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

   if 16000 < 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

      \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)} \]
   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 75.1%

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

   5. Simplified 75.1%

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

   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 65.6%

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

   8. Simplified 65.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 16000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.5% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 7.5e+48) x (* x (- 1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.5e+48) {
		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 <= 7.5d+48) 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 <= 7.5e+48) {
		tmp = x;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 7.5e+48:
		tmp = x
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7.5e+48)
		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 <= 7.5e+48)
		tmp = x;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 7.5e+48], x, N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000006e48

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 67.8%

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

   if 7.5000000000000006e48 < y

   1. Initial program 82.2%

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

   3. Taylor expanded in 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 79.7%

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

   5. Simplified 79.7%

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

   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 60.5%

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

   8. Simplified 60.5%

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

Alternative 10: 59.3% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 63.5%

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

Developer Target 1: 97.3% 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 2024155 
(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))))))