SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 94.3% → 98.0%

Time: 9.6s

Alternatives: 9

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: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.75 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\tanh \left(\frac{t}{y\_m}\right) - \tanh \left(\frac{x}{y\_m}\right)\right), y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 1.75e+175)
   (fma (* z (- (tanh (/ t y_m)) (tanh (/ x y_m)))) y_m x)
   (fma (- t x) z x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 1.75e+175) {
		tmp = fma((z * (tanh((t / y_m)) - tanh((x / y_m)))), y_m, x);
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 1.75e+175)
		tmp = fma(Float64(z * Float64(tanh(Float64(t / y_m)) - tanh(Float64(x / y_m)))), y_m, x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 1.75e+175], N[(N[(z * N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$95$m + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7500000000000002e175

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 1.7500000000000002e175 < y

   1. Initial program 59.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\tanh \left(\frac{t}{y\_m}\right) - \frac{x}{y\_m}, z \cdot y\_m, x\right)\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y\_m} - \tanh \left(\frac{x}{y\_m}\right)\right) \cdot z, y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (fma (- (tanh (/ t y_m)) (/ x y_m)) (* z y_m) x)))
   (if (<= t -1.55e+47)
     t_1
     (if (<= t 4.6e+49)
       (fma (* (- (/ t y_m) (tanh (/ x y_m))) z) y_m x)
       t_1))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = fma((tanh((t / y_m)) - (x / y_m)), (z * y_m), x);
	double tmp;
	if (t <= -1.55e+47) {
		tmp = t_1;
	} else if (t <= 4.6e+49) {
		tmp = fma((((t / y_m) - tanh((x / y_m))) * z), y_m, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = fma(Float64(tanh(Float64(t / y_m)) - Float64(x / y_m)), Float64(z * y_m), x)
	tmp = 0.0
	if (t <= -1.55e+47)
		tmp = t_1;
	elseif (t <= 4.6e+49)
		tmp = fma(Float64(Float64(Float64(t / y_m) - tanh(Float64(x / y_m))) * z), y_m, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] - N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.55e+47], t$95$1, If[LessEqual[t, 4.6e+49], N[(N[(N[(N[(t / y$95$m), $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y$95$m + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.55e47 or 4.60000000000000004e49 < t

   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 y around inf

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

      1. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 71.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

   if -1.55e47 < t < 4.60000000000000004e49

   1. Initial program 85.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 85.2

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

   7. Applied rewrites 85.2%

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

Alternative 3: 68.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.5 \cdot 10^{-187}:\\ \;\;\;\;z \cdot t + x\\ \mathbf{elif}\;y\_m \leq 5.5 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y\_m} - \tanh \left(\frac{x}{y\_m}\right)\right) \cdot z, y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 1.5e-187)
   (+ (* z t) x)
   (if (<= y_m 5.5e+174)
     (fma (* (- (/ t y_m) (tanh (/ x y_m))) z) y_m x)
     (fma (- t x) z x))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 1.5e-187) {
		tmp = (z * t) + x;
	} else if (y_m <= 5.5e+174) {
		tmp = fma((((t / y_m) - tanh((x / y_m))) * z), y_m, x);
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 1.5e-187)
		tmp = Float64(Float64(z * t) + x);
	elseif (y_m <= 5.5e+174)
		tmp = fma(Float64(Float64(Float64(t / y_m) - tanh(Float64(x / y_m))) * z), y_m, x);
	else
		tmp = fma(Float64(t - x), z, 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-187], N[(N[(z * t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y$95$m, 5.5e+174], N[(N[(N[(N[(t / y$95$m), $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y$95$m + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.50000000000000002e-187

   1. Initial program 92.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 59.6%

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

   if 1.50000000000000002e-187 < y < 5.4999999999999998e174

   1. Initial program 96.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 71.2

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

   7. Applied rewrites 71.2%

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

   if 5.4999999999999998e174 < y

   1. Initial program 59.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-187}:\\ \;\;\;\;z \cdot t + x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ \mathbf{if}\;z \leq -1.6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (* (- t x) z)))
   (if (<= z -1.6) t_1 (if (<= z 1.75e-17) (fma (- x) z x) t_1))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (t - x) * z;
	double tmp;
	if (z <= -1.6) {
		tmp = t_1;
	} else if (z <= 1.75e-17) {
		tmp = fma(-x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(t - x) * z)
	tmp = 0.0
	if (z <= -1.6)
		tmp = t_1;
	elseif (z <= 1.75e-17)
		tmp = fma(Float64(-x), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.6], t$95$1, If[LessEqual[z, 1.75e-17], N[((-x) * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6000000000000001 or 1.7500000000000001e-17 < z

   1. Initial program 82.7%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 45.8%

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

   if -1.6000000000000001 < z < 1.7500000000000001e-17

   1. Initial program 98.8%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 86.7%

      \[\leadsto \mathsf{fma}\left(-x, z, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 20.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-165}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-68}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= t -1.35e-165) (* z t) (if (<= t 3e-68) (* (- x) z) (* z t))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (t <= -1.35e-165) {
		tmp = z * t;
	} else if (t <= 3e-68) {
		tmp = -x * z;
	} 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 (t <= (-1.35d-165)) then
        tmp = z * t
    else if (t <= 3d-68) then
        tmp = -x * z
    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 (t <= -1.35e-165) {
		tmp = z * t;
	} else if (t <= 3e-68) {
		tmp = -x * z;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if t <= -1.35e-165:
		tmp = z * t
	elif t <= 3e-68:
		tmp = -x * z
	else:
		tmp = z * t
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (t <= -1.35e-165)
		tmp = Float64(z * t);
	elseif (t <= 3e-68)
		tmp = Float64(Float64(-x) * z);
	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 (t <= -1.35e-165)
		tmp = z * t;
	elseif (t <= 3e-68)
		tmp = -x * z;
	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[t, -1.35e-165], N[(z * t), $MachinePrecision], If[LessEqual[t, 3e-68], N[((-x) * z), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3499999999999999e-165 or 3e-68 < t

   1. Initial program 91.8%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 24.4%

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

   if -1.3499999999999999e-165 < t < 3e-68

   1. Initial program 87.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 36.8%

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

   9. Taylor expanded in t around 0

      \[\leadsto \left(-1 \cdot x\right) \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 30.0%

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

Alternative 6: 64.1% accurate, 14.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.55 \cdot 10^{-124}:\\ \;\;\;\;z \cdot t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 3.55e-124) (+ (* z t) x) (fma (- t x) z x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 3.55e-124) {
		tmp = (z * t) + x;
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 3.55e-124)
		tmp = Float64(Float64(z * t) + x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 3.55e-124], N[(N[(z * t), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.55000000000000019e-124

   1. Initial program 93.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 59.2%

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

   if 3.55000000000000019e-124 < y

   1. Initial program 83.4%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 72.5

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

   5. Applied rewrites 72.5%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.55 \cdot 10^{-124}:\\ \;\;\;\;z \cdot t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+26}:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= z -7.8e+26) (* (- t x) z) (+ (* z t) x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (z <= -7.8e+26) {
		tmp = (t - x) * z;
	} 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 (z <= (-7.8d+26)) then
        tmp = (t - x) * z
    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 (z <= -7.8e+26) {
		tmp = (t - x) * z;
	} else {
		tmp = (z * t) + x;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if z <= -7.8e+26:
		tmp = (t - x) * z
	else:
		tmp = (z * t) + x
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (z <= -7.8e+26)
		tmp = Float64(Float64(t - x) * z);
	else
		tmp = Float64(Float64(z * t) + x);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (z <= -7.8e+26)
		tmp = (t - x) * z;
	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[z, -7.8e+26], N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision], N[(N[(z * t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.8e26

   1. Initial program 78.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 59.1%

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

   if -7.8e26 < z

   1. Initial program 93.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 64.9

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

   5. Applied rewrites 64.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 66.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+26}:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 25.7% accurate, 26.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \left(t - x\right) \cdot z \end{array} \]

FPCore

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

C

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

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 = (t - x) * z
end function

Java

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

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	return Float64(Float64(t - x) * z)
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 63.6

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

5. Applied rewrites 63.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 30.5%

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

Alternative 9: 16.7% accurate, 39.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ z \cdot t \end{array} \]

FPCore

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

C

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

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 = z * t
end function

Java

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

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	return Float64(z * t)
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(z * t), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 63.6

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

5. Applied rewrites 63.6%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 19.3%

   \[\leadsto z \cdot \color{blue}{t} \]
9. Add Preprocessing

Developer Target 1: 97.6% 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 2024276 
(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))))))