SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.4% → 98.4%

Time: 8.6s

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: 93.4% 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, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.72 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y\_m}\right) - \tanh \left(\frac{x}{y\_m}\right)\right) \cdot z, y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 1.72e+187)
   (fma (* (- (tanh (/ 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 <= 1.72e+187) {
		tmp = fma(((tanh((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 <= 1.72e+187)
		tmp = fma(Float64(Float64(tanh(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, 1.72e+187], N[(N[(N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $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 2 regimes

2. if y < 1.72e187

   1. Initial program 94.7%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 97.2

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

   4. Applied rewrites 97.2%

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

   if 1.72e187 < y

   1. Initial program 75.9%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 2: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-59} \lor \neg \left(x \leq 6.2 \cdot 10^{+84}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y\_m} - \tanh \left(\frac{x}{y\_m}\right), z \cdot y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y\_m}\right) - \frac{x}{y\_m}\right) \cdot z, y\_m, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (or (<= x -6.8e-59) (not (<= x 6.2e+84)))
   (fma (- (/ t y_m) (tanh (/ x y_m))) (* z y_m) x)
   (fma (* (- (tanh (/ t y_m)) (/ x y_m)) z) y_m x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if ((x <= -6.8e-59) || !(x <= 6.2e+84)) {
		tmp = fma(((t / y_m) - tanh((x / y_m))), (z * y_m), x);
	} else {
		tmp = fma(((tanh((t / y_m)) - (x / y_m)) * z), y_m, x);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if ((x <= -6.8e-59) || !(x <= 6.2e+84))
		tmp = fma(Float64(Float64(t / y_m) - tanh(Float64(x / y_m))), Float64(z * y_m), x);
	else
		tmp = fma(Float64(Float64(tanh(Float64(t / y_m)) - Float64(x / y_m)) * z), y_m, x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[Or[LessEqual[x, -6.8e-59], N[Not[LessEqual[x, 6.2e+84]], $MachinePrecision]], N[(N[(N[(t / y$95$m), $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] - N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y$95$m + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.80000000000000035e-59 or 6.20000000000000006e84 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 82.9

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

   5. Applied rewrites 82.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 82.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if -6.80000000000000035e-59 < x < 6.20000000000000006e84

   1. Initial program 89.3%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   2. Add Preprocessing
   3. 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 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 84.9

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

   7. Applied rewrites 84.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-59} \lor \neg \left(x \leq 6.2 \cdot 10^{+84}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right), z \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right) \cdot z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.65 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{elif}\;y\_m \leq 1.12 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y\_m}\right) - \frac{x}{y\_m}\right) \cdot z, y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 1.65e-112)
   (* (/ x z) z)
   (if (<= y_m 1.12e+185)
     (fma (* (- (tanh (/ t y_m)) (/ 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 <= 1.65e-112) {
		tmp = (x / z) * z;
	} else if (y_m <= 1.12e+185) {
		tmp = fma(((tanh((t / y_m)) - (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 <= 1.65e-112)
		tmp = Float64(Float64(x / z) * z);
	elseif (y_m <= 1.12e+185)
		tmp = fma(Float64(Float64(tanh(Float64(t / y_m)) - 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, 1.65e-112], N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y$95$m, 1.12e+185], N[(N[(N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] - N[(x / y$95$m), $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 < 1.65e-112

   1. Initial program 94.5%

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

   3. 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 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 50.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 27.1%

       \[\leadsto \left(t - x\right) \cdot z \]

   12. Taylor expanded in z around 0

       \[\leadsto \frac{x}{z} \cdot z \]
   13. Step-by-step derivation

   14. Applied rewrites 49.4%

       \[\leadsto \frac{x}{z} \cdot z \]

   if 1.65e-112 < y < 1.11999999999999996e185

   1. Initial program 95.5%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   2. 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 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 79.8

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

   7. Applied rewrites 79.8%

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

   if 1.11999999999999996e185 < y

   1. Initial program 75.9%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 4: 71.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 6.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(z, t, z \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 6.4e-53) (* (/ x z) z) (+ x (fma z t (* z (- x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 6.4e-53) {
		tmp = (x / z) * z;
	} else {
		tmp = x + fma(z, t, (z * -x));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 6.4e-53)
		tmp = Float64(Float64(x / z) * z);
	else
		tmp = Float64(x + fma(z, t, Float64(z * Float64(-x))));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 6.4e-53], N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision], N[(x + N[(z * t + N[(z * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4000000000000002e-53

   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 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 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 25.2%

       \[\leadsto \left(t - x\right) \cdot z \]

   12. Taylor expanded in z around 0

       \[\leadsto \frac{x}{z} \cdot z \]
   13. Step-by-step derivation

   14. Applied rewrites 49.7%

       \[\leadsto \frac{x}{z} \cdot z \]

   if 6.4000000000000002e-53 < y

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{z \cdot \left(t - x\right)} \]
   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 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 80.9%

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

Alternative 5: 72.1% accurate, 10.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 6.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 6.4e-53) (* (/ x z) z) (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 <= 6.4e-53) {
		tmp = (x / z) * z;
	} 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 <= 6.4e-53)
		tmp = Float64(Float64(x / z) * z);
	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, 6.4e-53], N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4000000000000002e-53

   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 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 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 25.2%

       \[\leadsto \left(t - x\right) \cdot z \]

   12. Taylor expanded in z around 0

       \[\leadsto \frac{x}{z} \cdot z \]
   13. Step-by-step derivation

   14. Applied rewrites 49.7%

       \[\leadsto \frac{x}{z} \cdot z \]

   if 6.4000000000000002e-53 < y

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)} \]
   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 80.9

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

   5. Applied rewrites 80.9%

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

Alternative 6: 62.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-32} \lor \neg \left(z \leq 0.003\right):\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (or (<= z -3.6e-32) (not (<= z 0.003))) (* (- t x) z) (* (- 1.0 z) x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if ((z <= -3.6e-32) || !(z <= 0.003)) {
		tmp = (t - x) * z;
	} else {
		tmp = (1.0 - z) * 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 <= (-3.6d-32)) .or. (.not. (z <= 0.003d0))) then
        tmp = (t - x) * z
    else
        tmp = (1.0d0 - z) * 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 <= -3.6e-32) || !(z <= 0.003)) {
		tmp = (t - x) * z;
	} else {
		tmp = (1.0 - z) * x;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if (z <= -3.6e-32) or not (z <= 0.003):
		tmp = (t - x) * z
	else:
		tmp = (1.0 - z) * x
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if ((z <= -3.6e-32) || !(z <= 0.003))
		tmp = Float64(Float64(t - x) * z);
	else
		tmp = Float64(Float64(1.0 - z) * x);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if ((z <= -3.6e-32) || ~((z <= 0.003)))
		tmp = (t - x) * z;
	else
		tmp = (1.0 - z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[Or[LessEqual[z, -3.6e-32], N[Not[LessEqual[z, 0.003]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999993e-32 or 0.0030000000000000001 < z

   1. Initial program 88.0%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 47.0

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

   5. Applied rewrites 47.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 46.6%

       \[\leadsto \left(t - x\right) \cdot z \]

   if -3.59999999999999993e-32 < z < 0.0030000000000000001

   1. Initial program 98.6%

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

   3. Taylor expanded in y around inf

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

      1. +-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 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 82.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-32} \lor \neg \left(z \leq 0.003\right):\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.1% accurate, 14.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 6.8 \cdot 10^{-103}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 6.8e-103) (* (- 1.0 z) 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 <= 6.8e-103) {
		tmp = (1.0 - z) * 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 <= 6.8e-103)
		tmp = Float64(Float64(1.0 - z) * 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, 6.8e-103], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.80000000000000006e-103

   1. Initial program 94.5%

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

   3. 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 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.5%

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

   if 6.80000000000000006e-103 < y

   1. Initial program 90.6%

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

   3. Taylor expanded in y around inf

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

      1. +-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 70.1

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

   5. Applied rewrites 70.1%

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

Alternative 8: 26.9% 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 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

   \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)} \]
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 62.6

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

5. Applied rewrites 62.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 52.6%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 29.6%

    \[\leadsto \left(t - x\right) \cdot z \]
12. Add Preprocessing

Alternative 9: 17.3% 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 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

   \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)} \]
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 62.6

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

5. Applied rewrites 62.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 22.8%

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

Developer Target 1: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024317 
(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))))))