SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.4% → 97.1%

Time: 14.4s

Alternatives: 12

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: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.1%

   \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
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. lower-*.f64 97.3

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

4. Applied rewrites 97.3%

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

Alternative 2: 69.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ t_2 := x + \left(z \cdot y\right) \cdot \frac{1}{\frac{y + \frac{\frac{y \cdot \left(x \cdot x\right)}{t}}{t}}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{elif}\;t\_1 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+30}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(t, \frac{y}{x}, y\right)}{-x}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+281}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (* (- (tanh (/ t y)) (tanh (/ x y))) (* z y))))
        (t_2 (+ x (* (* z y) (/ 1.0 (/ (+ y (/ (/ (* y (* x x)) t) t)) t))))))
   (if (<= t_1 (- INFINITY))
     (* z (- t x))
     (if (<= t_1 -10000000.0)
       t_2
       (if (<= t_1 5e+30)
         (+ x (* (* z y) (/ 1.0 (/ (fma t (/ y x) y) (- x)))))
         (if (<= t_1 2e+281) t_2 (fma z (- t x) x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + ((tanh((t / y)) - tanh((x / y))) * (z * y));
	double t_2 = x + ((z * y) * (1.0 / ((y + (((y * (x * x)) / t) / t)) / t)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z * (t - x);
	} else if (t_1 <= -10000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+30) {
		tmp = x + ((z * y) * (1.0 / (fma(t, (y / x), y) / -x)));
	} else if (t_1 <= 2e+281) {
		tmp = t_2;
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))) * Float64(z * y)))
	t_2 = Float64(x + Float64(Float64(z * y) * Float64(1.0 / Float64(Float64(y + Float64(Float64(Float64(y * Float64(x * x)) / t) / t)) / t))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z * Float64(t - x));
	elseif (t_1 <= -10000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e+30)
		tmp = Float64(x + Float64(Float64(z * y) * Float64(1.0 / Float64(fma(t, Float64(y / x), y) / Float64(-x)))));
	elseif (t_1 <= 2e+281)
		tmp = t_2;
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(z * y), $MachinePrecision] * N[(1.0 / N[(N[(y + N[(N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -10000000.0], t$95$2, If[LessEqual[t$95$1, 5e+30], N[(x + N[(N[(z * y), $MachinePrecision] * N[(1.0 / N[(N[(t * N[(y / x), $MachinePrecision] + y), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+281], t$95$2, N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 52.8%

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

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -1e7 or 4.9999999999999998e30 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 2.0000000000000001e281

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 46.1

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

   5. Applied rewrites 46.1%

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

   7. Applied rewrites 46.1%

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

   8. Taylor expanded in t around -inf

      \[\leadsto x + \left(y \cdot z\right) \cdot \frac{1}{-1 \cdot \color{blue}{\frac{-1 \cdot y + -1 \cdot \frac{x \cdot y + \frac{{x}^{2} \cdot y}{t}}{t}}{t}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

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

   11. Taylor expanded in x around inf

       \[\leadsto x + \left(y \cdot z\right) \cdot \frac{1}{\frac{\left(\mathsf{neg}\left(y\right)\right) - \frac{\frac{{x}^{2} \cdot y}{t}}{t}}{\mathsf{neg}\left(t\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 74.4%

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

   if -1e7 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 4.9999999999999998e30

   1. Initial program 97.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 55.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 70.0%

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

   if 2.0000000000000001e281 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 52.4%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{elif}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -10000000:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \frac{1}{\frac{y + \frac{\frac{y \cdot \left(x \cdot x\right)}{t}}{t}}{t}}\\ \mathbf{elif}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 5 \cdot 10^{+30}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(t, \frac{y}{x}, y\right)}{-x}}\\ \mathbf{elif}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{+281}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \frac{1}{\frac{y + \frac{\frac{y \cdot \left(x \cdot x\right)}{t}}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y \cdot \tanh \left(\frac{t}{y}\right), z, \mathsf{fma}\left(x, -z, x\right)\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-231}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (* y (tanh (/ t y))) z (fma x (- z) x))))
   (if (<= t -1.8e+33)
     t_1
     (if (<= t 2.05e-231) (fma (* z (- (/ t y) (tanh (/ x y)))) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y * tanh((t / y))), z, fma(x, -z, x));
	double tmp;
	if (t <= -1.8e+33) {
		tmp = t_1;
	} else if (t <= 2.05e-231) {
		tmp = fma((z * ((t / y) - tanh((x / y)))), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y * tanh(Float64(t / y))), z, fma(x, Float64(-z), x))
	tmp = 0.0
	if (t <= -1.8e+33)
		tmp = t_1;
	elseif (t <= 2.05e-231)
		tmp = fma(Float64(z * Float64(Float64(t / y) - tanh(Float64(x / y)))), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z + N[(x * (-z) + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.8e+33], t$95$1, If[LessEqual[t, 2.05e-231], N[(N[(z * N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8000000000000001e33 or 2.0500000000000001e-231 < t

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 89.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 83.1

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

   9. Applied rewrites 83.1%

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

   if -1.8000000000000001e33 < t < 2.0500000000000001e-231

   1. Initial program 89.8%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   2. Add Preprocessing
   3. 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. lower-*.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 91.6

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

   7. Applied rewrites 91.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \tanh \left(\frac{t}{y}\right), z, \mathsf{fma}\left(x, -z, x\right)\right)\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-231}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \tanh \left(\frac{t}{y}\right), z, \mathsf{fma}\left(x, -z, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{\frac{y + \frac{x \cdot \mathsf{fma}\left(y, \frac{x}{t}, y\right)}{t}}{t}}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \tanh \left(\frac{t}{y}\right), z, \mathsf{fma}\left(x, -z, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.7e+119)
   (fma (* y (/ 1.0 (/ (+ y (/ (* x (fma y (/ x t) y)) t)) t))) z x)
   (fma (* y (tanh (/ t y))) z (fma x (- z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.7e+119) {
		tmp = fma((y * (1.0 / ((y + ((x * fma(y, (x / t), y)) / t)) / t))), z, x);
	} else {
		tmp = fma((y * tanh((t / y))), z, fma(x, -z, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.7e+119)
		tmp = fma(Float64(y * Float64(1.0 / Float64(Float64(y + Float64(Float64(x * fma(y, Float64(x / t), y)) / t)) / t))), z, x);
	else
		tmp = fma(Float64(y * tanh(Float64(t / y))), z, fma(x, Float64(-z), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.7e+119], N[(N[(y * N[(1.0 / N[(N[(y + N[(N[(x * N[(y * N[(x / t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z + N[(x * (-z) + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6999999999999998e119

   1. Initial program 95.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 48.4

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

   5. Applied rewrites 48.4%

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in t around -inf

      \[\leadsto x + \left(y \cdot z\right) \cdot \frac{1}{-1 \cdot \color{blue}{\frac{-1 \cdot y + -1 \cdot \frac{x \cdot y + \frac{{x}^{2} \cdot y}{t}}{t}}{t}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.0%

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

   12. Applied rewrites 83.3%

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

   if -2.6999999999999998e119 < x

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 88.3%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 83.9

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

   9. Applied rewrites 83.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{\frac{y + \frac{x \cdot \mathsf{fma}\left(y, \frac{x}{t}, y\right)}{t}}{t}}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \tanh \left(\frac{t}{y}\right), z, \mathsf{fma}\left(x, -z, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x \cdot \left(-x\right)}{t + x}, x\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-104}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(x, \frac{y}{t}, y\right)}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3e-224)
   (fma z (/ (* x (- x)) (+ t x)) x)
   (if (<= y 2.2e-104)
     (+ x (* (* z y) (/ 1.0 (/ (fma x (/ y t) y) t))))
     (fma z (- t x) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3e-224) {
		tmp = fma(z, ((x * -x) / (t + x)), x);
	} else if (y <= 2.2e-104) {
		tmp = x + ((z * y) * (1.0 / (fma(x, (y / t), y) / t)));
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3e-224)
		tmp = fma(z, Float64(Float64(x * Float64(-x)) / Float64(t + x)), x);
	elseif (y <= 2.2e-104)
		tmp = Float64(x + Float64(Float64(z * y) * Float64(1.0 / Float64(fma(x, Float64(y / t), y) / t))));
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3e-224], N[(z * N[(N[(x * (-x)), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.2e-104], N[(x + N[(N[(z * y), $MachinePrecision] * N[(1.0 / N[(N[(x * N[(y / t), $MachinePrecision] + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.99999999999999982e-224

   1. Initial program 90.9%

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

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

      3. lower--.f64 64.5

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

   5. Applied rewrites 64.5%

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

   7. Applied rewrites 46.0%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(z, \frac{-1 \cdot {x}^{2}}{\color{blue}{t} + x}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 47.2%

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

   if 2.99999999999999982e-224 < y < 2.20000000000000012e-104

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 30.5

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

   5. Applied rewrites 30.5%

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

   7. Applied rewrites 30.5%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 54.6%

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

   if 2.20000000000000012e-104 < y

   1. Initial program 91.6%

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

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

      3. lower--.f64 71.1

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

   5. Applied rewrites 71.1%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x \cdot \left(-x\right)}{t + x}, x\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-104}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(x, \frac{y}{t}, y\right)}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x \cdot \left(-x\right)}{t + x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.6e-128) (fma z (/ (* x (- x)) (+ t x)) x) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.6e-128) {
		tmp = fma(z, ((x * -x) / (t + x)), x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.6e-128)
		tmp = fma(z, Float64(Float64(x * Float64(-x)) / Float64(t + x)), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 6.6e-128], N[(z * N[(N[(x * (-x)), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.6e-128

   1. Initial program 92.1%

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

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

      3. lower--.f64 62.9

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

   5. Applied rewrites 62.9%

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

   7. Applied rewrites 43.9%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(z, \frac{-1 \cdot {x}^{2}}{\color{blue}{t} + x}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 47.9%

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

   if 6.6e-128 < y

   1. Initial program 92.2%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 69.7

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

   5. Applied rewrites 69.7%

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

Alternative 7: 57.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t \cdot t}{t + x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5.5e-124) (fma z (/ (* t t) (+ t x)) x) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.5e-124) {
		tmp = fma(z, ((t * t) / (t + x)), x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.5e-124)
		tmp = fma(z, Float64(Float64(t * t) / Float64(t + x)), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.5e-124], N[(z * N[(N[(t * t), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.50000000000000016e-124

   1. Initial program 92.1%

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

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

      3. lower--.f64 62.2

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

   5. Applied rewrites 62.2%

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{{t}^{2}}{\color{blue}{t} + x}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 56.2%

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

   if 5.50000000000000016e-124 < y

   1. Initial program 92.0%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 71.3

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

   5. Applied rewrites 71.3%

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

Alternative 8: 62.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(t - x\right)\\ \mathbf{if}\;z \leq -5500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- t x))))
   (if (<= z -5500000000000.0) t_1 (if (<= z 2.9e-6) (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 <= -5500000000000.0) {
		tmp = t_1;
	} else if (z <= 2.9e-6) {
		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 <= -5500000000000.0)
		tmp = t_1;
	elseif (z <= 2.9e-6)
		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, -5500000000000.0], t$95$1, If[LessEqual[z, 2.9e-6], N[(z * (-x) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e12 or 2.9000000000000002e-6 < z

   1. Initial program 82.8%

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

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

      3. lower--.f64 46.5

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 45.6%

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

   if -5.5e12 < z < 2.9000000000000002e-6

   1. Initial program 99.3%

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

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

      3. lower--.f64 79.5

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

   5. Applied rewrites 79.5%

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

   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 83.1%

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

Alternative 9: 58.7% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(z, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.45e-102) (fma z (- x) x) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.45e-102) {
		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 <= 2.45e-102)
		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, 2.45e-102], N[(z * (-x) + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4499999999999999e-102

   1. Initial program 92.3%

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

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

      3. lower--.f64 62.5

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

   5. Applied rewrites 62.5%

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

   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 55.6%

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

   if 2.4499999999999999e-102 < y

   1. Initial program 91.6%

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

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

      3. lower--.f64 71.1

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

   5. Applied rewrites 71.1%

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

Alternative 10: 18.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-155}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.2e-155) (* z (- x)) (* z t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.2e-155) {
		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 (x <= (-6.2d-155)) 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 (x <= -6.2e-155) {
		tmp = z * -x;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -6.2e-155:
		tmp = z * -x
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.2e-155)
		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 (x <= -6.2e-155)
		tmp = z * -x;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6.2e-155], N[(z * (-x)), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e-155

   1. Initial program 90.4%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 22.3%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 15.9%

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

   if -6.2e-155 < x

   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

      \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)} \]
   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 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 24.9%

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

Alternative 11: 26.8% 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 92.1%

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

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

   3. lower--.f64 65.2

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

5. Applied rewrites 65.2%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 26.4%

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

Alternative 12: 17.2% 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 92.1%

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

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

   3. lower--.f64 65.2

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

5. Applied rewrites 65.2%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 18.4%

   \[\leadsto z \cdot \color{blue}{t} \]
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 2024221 
(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))))))