SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.6% → 98.4%

Time: 15.0s

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: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(y_m, z \cdot \left(\tanh \left(\frac{t}{y_m}\right) - \tanh \left(\frac{x}{y_m}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 1.5e+217)
   (fma y_m (* z (- (tanh (/ t y_m)) (tanh (/ x y_m)))) x)
   (+ x (* z (- t x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 1.5e+217) {
		tmp = fma(y_m, (z * (tanh((t / y_m)) - tanh((x / y_m)))), x);
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 1.5e+217)
		tmp = fma(y_m, Float64(z * Float64(tanh(Float64(t / y_m)) - tanh(Float64(x / y_m)))), x);
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 1.5e+217], N[(y$95$m * N[(z * N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.49999999999999988e217

   1. Initial program 94.3%

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

      1. +-commutative 94.3%

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

      2. associate-*l* 97.1%

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

      3. fma-def 97.1%

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

   3. Simplified 97.1%

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

   if 1.49999999999999988e217 < y

   1. Initial program 81.1%

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

   2. Taylor expanded in y around inf 100.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 2: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 8.6 \cdot 10^{+140}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y_m}\right) - \tanh \left(\frac{x}{y_m}\right)\right) \cdot \left(y_m \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 8.6e+140)
   (+ x (* (- (tanh (/ t y_m)) (tanh (/ x y_m))) (* y_m z)))
   (+ x (* z (- t x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 8.6e+140) {
		tmp = x + ((tanh((t / y_m)) - tanh((x / y_m))) * (y_m * z));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y_m <= 8.6d+140) then
        tmp = x + ((tanh((t / y_m)) - tanh((x / y_m))) * (y_m * z))
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 8.6e+140) {
		tmp = x + ((Math.tanh((t / y_m)) - Math.tanh((x / y_m))) * (y_m * z));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if y_m <= 8.6e+140:
		tmp = x + ((math.tanh((t / y_m)) - math.tanh((x / y_m))) * (y_m * z))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 8.6e+140)
		tmp = Float64(x + Float64(Float64(tanh(Float64(t / y_m)) - tanh(Float64(x / y_m))) * Float64(y_m * z)));
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (y_m <= 8.6e+140)
		tmp = x + ((tanh((t / y_m)) - tanh((x / y_m))) * (y_m * z));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 8.6e+140], N[(x + N[(N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.60000000000000004e140

   1. Initial program 95.6%

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

   if 8.60000000000000004e140 < y

   1. Initial program 78.4%

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

   2. Taylor expanded in y around inf 93.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{+140}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 3: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y_m}\right)\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y_m, z \cdot t_1, x\right)\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;x + z \cdot \left(y_m \cdot \left(\frac{t}{y_m} - \tanh \left(\frac{x}{y_m}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_1 \cdot \left(y_m \cdot z\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y_m))))
   (if (<= t -1.9e-19)
     (fma y_m (* z t_1) x)
     (if (<= t 6.4e-24)
       (+ x (* z (* y_m (- (/ t y_m) (tanh (/ x y_m))))))
       (+ x (* t_1 (* y_m z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = tanh((t / y_m));
	double tmp;
	if (t <= -1.9e-19) {
		tmp = fma(y_m, (z * t_1), x);
	} else if (t <= 6.4e-24) {
		tmp = x + (z * (y_m * ((t / y_m) - tanh((x / y_m)))));
	} else {
		tmp = x + (t_1 * (y_m * z));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = tanh(Float64(t / y_m))
	tmp = 0.0
	if (t <= -1.9e-19)
		tmp = fma(y_m, Float64(z * t_1), x);
	elseif (t <= 6.4e-24)
		tmp = Float64(x + Float64(z * Float64(y_m * Float64(Float64(t / y_m) - tanh(Float64(x / y_m))))));
	else
		tmp = Float64(x + Float64(t_1 * Float64(y_m * z)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.9e-19], N[(y$95$m * N[(z * t$95$1), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 6.4e-24], N[(x + N[(z * N[(y$95$m * N[(N[(t / y$95$m), $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$1 * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9e-19

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-*l* 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around 0 5.1%

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

      1. associate-/r* 5.1%

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

      2. div-sub 5.1%

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

      3. rec-exp 5.2%

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

      4. rec-exp 5.2%

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

      5. tanh-def-a 87.8%

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

   6. Simplified 87.8%

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

   if -1.9e-19 < t < 6.40000000000000025e-24

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. associate-*l* 94.6%

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

      3. fma-def 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in t around 0 90.2%

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

      1. fma-udef 90.2%

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

      2. associate-*l* 84.9%

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

      3. *-commutative 84.9%

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

      4. associate-*l* 91.7%

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

   6. Applied egg-rr 91.7%

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

   if 6.40000000000000025e-24 < t

   1. Initial program 98.3%

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

   2. Taylor expanded in x around 0 10.6%

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

      1. associate-*r* 10.5%

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

      2. associate-/r* 10.5%

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

      3. div-sub 10.5%

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

      4. rec-exp 10.5%

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

      5. rec-exp 10.5%

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

      6. tanh-def-a 88.3%

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

   4. Simplified 88.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 4: 86.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-18} \lor \neg \left(t \leq 1.85 \cdot 10^{-24}\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y_m}\right) \cdot \left(y_m \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y_m \cdot \left(\frac{t}{y_m} - \tanh \left(\frac{x}{y_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (or (<= t -1.45e-18) (not (<= t 1.85e-24)))
   (+ x (* (tanh (/ t y_m)) (* y_m z)))
   (+ x (* z (* y_m (- (/ t y_m) (tanh (/ x y_m))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if ((t <= -1.45e-18) || !(t <= 1.85e-24)) {
		tmp = x + (tanh((t / y_m)) * (y_m * z));
	} else {
		tmp = x + (z * (y_m * ((t / y_m) - tanh((x / y_m)))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.45d-18)) .or. (.not. (t <= 1.85d-24))) then
        tmp = x + (tanh((t / y_m)) * (y_m * z))
    else
        tmp = x + (z * (y_m * ((t / y_m) - tanh((x / y_m)))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if ((t <= -1.45e-18) || !(t <= 1.85e-24)) {
		tmp = x + (Math.tanh((t / y_m)) * (y_m * z));
	} else {
		tmp = x + (z * (y_m * ((t / y_m) - Math.tanh((x / y_m)))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if (t <= -1.45e-18) or not (t <= 1.85e-24):
		tmp = x + (math.tanh((t / y_m)) * (y_m * z))
	else:
		tmp = x + (z * (y_m * ((t / y_m) - math.tanh((x / y_m)))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if ((t <= -1.45e-18) || !(t <= 1.85e-24))
		tmp = Float64(x + Float64(tanh(Float64(t / y_m)) * Float64(y_m * z)));
	else
		tmp = Float64(x + Float64(z * Float64(y_m * Float64(Float64(t / y_m) - tanh(Float64(x / y_m))))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if ((t <= -1.45e-18) || ~((t <= 1.85e-24)))
		tmp = x + (tanh((t / y_m)) * (y_m * z));
	else
		tmp = x + (z * (y_m * ((t / y_m) - tanh((x / y_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[Or[LessEqual[t, -1.45e-18], N[Not[LessEqual[t, 1.85e-24]], $MachinePrecision]], N[(x + N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y$95$m * N[(N[(t / y$95$m), $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.45e-18 or 1.8499999999999999e-24 < t

   1. Initial program 98.4%

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

   2. Taylor expanded in x around 0 7.6%

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

      1. associate-*r* 7.5%

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

      2. associate-/r* 7.5%

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

      3. div-sub 7.5%

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

      4. rec-exp 7.6%

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

      5. rec-exp 7.6%

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

      6. tanh-def-a 88.1%

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

   4. Simplified 88.1%

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

   if -1.45e-18 < t < 1.8499999999999999e-24

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. associate-*l* 94.6%

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

      3. fma-def 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in t around 0 90.2%

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

      1. fma-udef 90.2%

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

      2. associate-*l* 84.9%

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

      3. *-commutative 84.9%

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

      4. associate-*l* 91.7%

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

   6. Applied egg-rr 91.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-18} \lor \neg \left(t \leq 1.85 \cdot 10^{-24}\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 5: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 4 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;y_m \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;\tanh \left(\frac{t}{y_m}\right) \cdot \left(y_m \cdot z\right)\\ \mathbf{elif}\;y_m \leq 5.5 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 4e-43)
   x
   (if (<= y_m 2.1e-15)
     (* (tanh (/ t y_m)) (* y_m z))
     (if (<= y_m 5.5e+39) x (+ x (* z (- t x)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 4e-43) {
		tmp = x;
	} else if (y_m <= 2.1e-15) {
		tmp = tanh((t / y_m)) * (y_m * z);
	} else if (y_m <= 5.5e+39) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y_m <= 4d-43) then
        tmp = x
    else if (y_m <= 2.1d-15) then
        tmp = tanh((t / y_m)) * (y_m * z)
    else if (y_m <= 5.5d+39) then
        tmp = x
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 4e-43) {
		tmp = x;
	} else if (y_m <= 2.1e-15) {
		tmp = Math.tanh((t / y_m)) * (y_m * z);
	} else if (y_m <= 5.5e+39) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if y_m <= 4e-43:
		tmp = x
	elif y_m <= 2.1e-15:
		tmp = math.tanh((t / y_m)) * (y_m * z)
	elif y_m <= 5.5e+39:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 4e-43)
		tmp = x;
	elseif (y_m <= 2.1e-15)
		tmp = Float64(tanh(Float64(t / y_m)) * Float64(y_m * z));
	elseif (y_m <= 5.5e+39)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (y_m <= 4e-43)
		tmp = x;
	elseif (y_m <= 2.1e-15)
		tmp = tanh((t / y_m)) * (y_m * z);
	elseif (y_m <= 5.5e+39)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 4e-43], x, If[LessEqual[y$95$m, 2.1e-15], N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5.5e+39], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.00000000000000031e-43 or 2.09999999999999981e-15 < y < 5.4999999999999997e39

   1. Initial program 95.1%

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

   2. Taylor expanded in x around inf 65.0%

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

   if 4.00000000000000031e-43 < y < 2.09999999999999981e-15

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*l* 99.8%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 0.6%

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

      1. associate-/r* 0.6%

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

      2. div-sub 0.6%

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

      3. rec-exp 0.6%

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

      4. rec-exp 0.6%

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

      5. tanh-def-a 80.1%

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

   6. Simplified 80.1%

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

   7. Taylor expanded in z around inf 0.7%

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

      1. associate-*r* 0.7%

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

      2. *-commutative 0.7%

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

      3. associate-*r/ 0.7%

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

      4. rec-exp 0.7%

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

      5. rec-exp 0.7%

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

      6. tanh-def-a 59.2%

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

      7. *-commutative 59.2%

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

   9. Simplified 59.2%

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

   if 5.4999999999999997e39 < 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. Taylor expanded in y around inf 88.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;\tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 6: 85.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 9.2 \cdot 10^{+129}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y_m}\right) \cdot \left(y_m \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 9.2e+129)
   (+ x (* (tanh (/ t y_m)) (* y_m z)))
   (+ x (* z (- t x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 9.2e+129) {
		tmp = x + (tanh((t / y_m)) * (y_m * z));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y_m <= 9.2d+129) then
        tmp = x + (tanh((t / y_m)) * (y_m * z))
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 9.2e+129) {
		tmp = x + (Math.tanh((t / y_m)) * (y_m * z));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if y_m <= 9.2e+129:
		tmp = x + (math.tanh((t / y_m)) * (y_m * z))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 9.2e+129)
		tmp = Float64(x + Float64(tanh(Float64(t / y_m)) * Float64(y_m * z)));
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (y_m <= 9.2e+129)
		tmp = x + (tanh((t / y_m)) * (y_m * z));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 9.2e+129], N[(x + N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.19999999999999961e129

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0 27.8%

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

      1. associate-*r* 27.6%

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

      2. associate-/r* 27.6%

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

      3. div-sub 27.6%

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

      4. rec-exp 27.6%

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

      5. rec-exp 27.6%

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

      6. tanh-def-a 79.8%

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

   4. Simplified 79.8%

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

   if 9.19999999999999961e129 < y

   1. Initial program 79.0%

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

   2. Taylor expanded in y around inf 91.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+129}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 7: 77.2% accurate, 23.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 1.35e+42) x (+ x (* z (- t x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 1.35e+42) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y_m <= 1.35d+42) then
        tmp = x
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 1.35e+42) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if y_m <= 1.35e+42:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 1.35e+42)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (y_m <= 1.35e+42)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 1.35e+42], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e42

   1. Initial program 95.3%

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

   2. Taylor expanded in x around inf 63.3%

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

   if 1.35e42 < 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. Taylor expanded in y around inf 88.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 8: 64.7% accurate, 30.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.45 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 1.45e+130) x (* z (- t x))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 1.45e+130) {
		tmp = x;
	} else {
		tmp = z * (t - x);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y_m <= 1.45d+130) then
        tmp = x
    else
        tmp = z * (t - x)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 1.45e+130) {
		tmp = x;
	} else {
		tmp = z * (t - x);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if y_m <= 1.45e+130:
		tmp = x
	else:
		tmp = z * (t - x)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 1.45e+130)
		tmp = x;
	else
		tmp = Float64(z * Float64(t - x));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (y_m <= 1.45e+130)
		tmp = x;
	else
		tmp = z * (t - x);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 1.45e+130], x, N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e130

   1. Initial program 95.6%

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

   2. Taylor expanded in x around inf 62.2%

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

   if 1.45e130 < y

   1. Initial program 79.0%

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

   2. Taylor expanded in y around inf 91.2%

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

   3. Taylor expanded in z around inf 70.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 9: 58.6% accurate, 42.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 2 \cdot 10^{+135}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t) :precision binary64 (if (<= y_m 2e+135) x (* z t)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 2e+135) {
		tmp = x;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y_m <= 2d+135) then
        tmp = x
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 2e+135) {
		tmp = x;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if y_m <= 2e+135:
		tmp = x
	else:
		tmp = z * t
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 2e+135)
		tmp = x;
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (y_m <= 2e+135)
		tmp = x;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 2e+135], x, N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999992e135

   1. Initial program 95.6%

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

   2. Taylor expanded in x around inf 62.2%

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

   if 1.99999999999999992e135 < y

   1. Initial program 79.0%

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

   2. Taylor expanded in y around inf 91.2%

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

   3. Taylor expanded in x around 0 48.6%

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

      1. *-commutative 48.6%

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

   5. Simplified 48.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+135}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 10: 60.0% accurate, 213.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t) :precision binary64 x)

C

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

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	return x;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	return x

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	return x
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := x

Derivation

1. Initial program 93.5%

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

2. Taylor expanded in x around inf 57.6%

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

3. Final simplification 57.6%

   \[\leadsto x \]

Developer target: 97.0% 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 2023326 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

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

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