SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.7% → 96.7%

Time: 13.6s

Alternatives: 8

Speedup: 0.5×

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: 93.7% 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: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\\ \mathbf{if}\;x + \left(y \cdot z\right) \cdot t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, y \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (tanh (/ t y)) (tanh (/ x y)))))
   (if (<= (+ x (* (* y z) t_1)) 2e+305) (fma t_1 (* y z) x) (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y)) - tanh((x / y));
	double tmp;
	if ((x + ((y * z) * t_1)) <= 2e+305) {
		tmp = fma(t_1, (y * z), x);
	} else {
		tmp = z * (t - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * z) * t_1)) <= 2e+305)
		tmp = fma(t_1, Float64(y * z), x);
	else
		tmp = Float64(z * Float64(t - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + N[(N[(y * z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2e+305], N[(t$95$1 * N[(y * z), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.9999999999999999e305

   1. Initial program 99.5%

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

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 1.9999999999999999e305 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 50.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 100.0%

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

Alternative 2: 75.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right), y \cdot z, x\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(z, \left(-x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(t, t, x \cdot \left(x + t\right)\right)}, x\right)\\ \mathbf{elif}\;x \leq 66:\\ \;\;\;\;\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}, y \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- (/ t y) (tanh (/ x y))) (* y z) x)))
   (if (<= x -1.4e+81)
     t_1
     (if (<= x -2.85e-21)
       (fma z (* (- (* x (* x x))) (/ 1.0 (fma t t (* x (+ x t))))) x)
       (if (<= x 66.0) (fma (- (tanh (/ t y)) (/ x y)) (* y z) x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(((t / y) - tanh((x / y))), (y * z), x);
	double tmp;
	if (x <= -1.4e+81) {
		tmp = t_1;
	} else if (x <= -2.85e-21) {
		tmp = fma(z, (-(x * (x * x)) * (1.0 / fma(t, t, (x * (x + t))))), x);
	} else if (x <= 66.0) {
		tmp = fma((tanh((t / y)) - (x / y)), (y * z), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(Float64(t / y) - tanh(Float64(x / y))), Float64(y * z), x)
	tmp = 0.0
	if (x <= -1.4e+81)
		tmp = t_1;
	elseif (x <= -2.85e-21)
		tmp = fma(z, Float64(Float64(-Float64(x * Float64(x * x))) * Float64(1.0 / fma(t, t, Float64(x * Float64(x + t))))), x);
	elseif (x <= 66.0)
		tmp = fma(Float64(tanh(Float64(t / y)) - Float64(x / y)), Float64(y * z), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -1.4e+81], t$95$1, If[LessEqual[x, -2.85e-21], N[(z * N[((-N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]) * N[(1.0 / N[(t * t + N[(x * N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 66.0], N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.39999999999999997e81 or 66 < x

   1. Initial program 99.0%

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

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

      5. lower-fma.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 70.5

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

   7. Applied rewrites 70.5%

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

   if -1.39999999999999997e81 < x < -2.8499999999999998e-21

   1. Initial program 96.2%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower--.f64 49.3

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

   5. Applied rewrites 49.3%

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 76.7%

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

   if -2.8499999999999998e-21 < x < 66

   1. Initial program 93.0%

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

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

      5. lower-fma.f64 93.0

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 80.9

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

   7. Applied rewrites 80.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right), y \cdot z, x\right)\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(z, \left(-x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(t, t, x \cdot \left(x + t\right)\right)}, x\right)\\ \mathbf{elif}\;x \leq 66:\\ \;\;\;\;\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}, y \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right), y \cdot z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{fma}\left(z, -x, x\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right), y \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.45e-145)
   (fma z (- x) x)
   (if (<= y 4.5e+181)
     (fma (- (/ t y) (tanh (/ x y))) (* y z) x)
     (fma z (- t x) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.45e-145) {
		tmp = fma(z, -x, x);
	} else if (y <= 4.5e+181) {
		tmp = fma(((t / y) - tanh((x / y))), (y * z), x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.45e-145)
		tmp = fma(z, Float64(-x), x);
	elseif (y <= 4.5e+181)
		tmp = fma(Float64(Float64(t / y) - tanh(Float64(x / y))), Float64(y * z), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.45e-145], N[(z * (-x) + x), $MachinePrecision], If[LessEqual[y, 4.5e+181], N[(N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.44999999999999992e-145

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 46.6%

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

   if 1.44999999999999992e-145 < y < 4.5e181

   1. Initial program 98.7%

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

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

      5. lower-fma.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 63.6

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

   7. Applied rewrites 63.6%

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

   if 4.5e181 < y

   1. Initial program 83.5%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 94.5

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

   5. Applied rewrites 94.5%

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

Alternative 4: 63.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(t - x\right)\\ \mathbf{if}\;z \leq -30.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(z, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- t x))))
   (if (<= z -30.5) t_1 (if (<= z 6.4e+21) (fma z (- x) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (t - x);
	double tmp;
	if (z <= -30.5) {
		tmp = t_1;
	} else if (z <= 6.4e+21) {
		tmp = fma(z, -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(t - x))
	tmp = 0.0
	if (z <= -30.5)
		tmp = t_1;
	elseif (z <= 6.4e+21)
		tmp = fma(z, Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -30.5], t$95$1, If[LessEqual[z, 6.4e+21], N[(z * (-x) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -30.5 or 6.4e21 < z

   1. Initial program 91.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 44.8%

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

   if -30.5 < z < 6.4e21

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 74.5%

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

Alternative 5: 20.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-165}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-163}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3e-165) (* z t) (if (<= t 4.8e-163) (* z (- x)) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e-165) {
		tmp = z * t;
	} else if (t <= 4.8e-163) {
		tmp = z * -x;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3d-165)) then
        tmp = z * t
    else if (t <= 4.8d-163) then
        tmp = z * -x
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e-165) {
		tmp = z * t;
	} else if (t <= 4.8e-163) {
		tmp = z * -x;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3e-165:
		tmp = z * t
	elif t <= 4.8e-163:
		tmp = z * -x
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3e-165)
		tmp = Float64(z * t);
	elseif (t <= 4.8e-163)
		tmp = Float64(z * Float64(-x));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3e-165)
		tmp = z * t;
	elseif (t <= 4.8e-163)
		tmp = z * -x;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3e-165], N[(z * t), $MachinePrecision], If[LessEqual[t, 4.8e-163], N[(z * (-x)), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.99999999999999979e-165 or 4.8000000000000001e-163 < t

   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. 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. lower-fma.f64 N/A

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

      3. lower--.f64 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 24.2%

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

   if -2.99999999999999979e-165 < t < 4.8000000000000001e-163

   1. Initial program 95.4%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 30.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 27.4%

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

Alternative 6: 59.3% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(z, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3e-120) (fma z (- x) x) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3e-120) {
		tmp = fma(z, -x, x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3e-120)
		tmp = fma(z, Float64(-x), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3e-120], N[(z * (-x) + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.00000000000000011e-120

   1. Initial program 95.4%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 46.2%

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

   if 3.00000000000000011e-120 < y

   1. Initial program 95.7%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 67.2

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

   5. Applied rewrites 67.2%

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

Alternative 7: 26.3% accurate, 26.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. 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. lower-fma.f64 N/A

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

   3. lower--.f64 56.5

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

5. Applied rewrites 56.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 29.2%

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

Alternative 8: 16.9% accurate, 39.8× speedup

Math

\[\begin{array}{l} \\ z \cdot t \end{array} \]

FPCore

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

C

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

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 = z * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(z * t), $MachinePrecision]

Derivation

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. 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. lower-fma.f64 N/A

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

   3. lower--.f64 56.5

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

5. Applied rewrites 56.5%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 19.6%

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

Developer Target 1: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(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))))))