SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.5% → 98.0%

Time: 11.2s

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: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.5 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y\_m}\right) - \tanh \left(\frac{x}{y\_m}\right)\right) \cdot y\_m, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 4.5e+176)
   (fma (* (- (tanh (/ t y_m)) (tanh (/ x y_m))) y_m) z x)
   (fma (- t x) z x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 4.5e+176) {
		tmp = fma(((tanh((t / y_m)) - tanh((x / y_m))) * y_m), z, x);
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 4.5e+176)
		tmp = fma(Float64(Float64(tanh(Float64(t / y_m)) - tanh(Float64(x / y_m))) * y_m), z, x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 4.5e+176], N[(N[(N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.50000000000000003e176

   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. 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. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if 4.50000000000000003e176 < y

   1. Initial program 69.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 79.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{1}{\frac{\frac{\mathsf{fma}\left(y\_m, x, \frac{\left(x \cdot x\right) \cdot y\_m}{t}\right)}{t} + y\_m}{t}} \cdot \left(z \cdot y\_m\right) + x\\ t_2 := \tanh \left(\frac{t}{y\_m}\right)\\ t_3 := x - \left(\tanh \left(\frac{x}{y\_m}\right) - t\_2\right) \cdot \left(z \cdot y\_m\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y\_m, t\_2, x\right) - z \cdot x\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1
         (+
          (*
           (/ 1.0 (/ (+ (/ (fma y_m x (/ (* (* x x) y_m) t)) t) y_m) t))
           (* z y_m))
          x))
        (t_2 (tanh (/ t y_m)))
        (t_3 (- x (* (- (tanh (/ x y_m)) t_2) (* z y_m)))))
   (if (<= t_3 -5e+298)
     (fma (- t x) z x)
     (if (<= t_3 -5e+173)
       t_1
       (if (<= t_3 2e+83)
         (- (fma (* z y_m) t_2 x) (* z x))
         (if (<= t_3 5e+304) t_1 (fma z t (* (- x) z))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = ((1.0 / (((fma(y_m, x, (((x * x) * y_m) / t)) / t) + y_m) / t)) * (z * y_m)) + x;
	double t_2 = tanh((t / y_m));
	double t_3 = x - ((tanh((x / y_m)) - t_2) * (z * y_m));
	double tmp;
	if (t_3 <= -5e+298) {
		tmp = fma((t - x), z, x);
	} else if (t_3 <= -5e+173) {
		tmp = t_1;
	} else if (t_3 <= 2e+83) {
		tmp = fma((z * y_m), t_2, x) - (z * x);
	} else if (t_3 <= 5e+304) {
		tmp = t_1;
	} else {
		tmp = fma(z, t, (-x * z));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y_m, x, Float64(Float64(Float64(x * x) * y_m) / t)) / t) + y_m) / t)) * Float64(z * y_m)) + x)
	t_2 = tanh(Float64(t / y_m))
	t_3 = Float64(x - Float64(Float64(tanh(Float64(x / y_m)) - t_2) * Float64(z * y_m)))
	tmp = 0.0
	if (t_3 <= -5e+298)
		tmp = fma(Float64(t - x), z, x);
	elseif (t_3 <= -5e+173)
		tmp = t_1;
	elseif (t_3 <= 2e+83)
		tmp = Float64(fma(Float64(z * y_m), t_2, x) - Float64(z * x));
	elseif (t_3 <= 5e+304)
		tmp = t_1;
	else
		tmp = fma(z, t, Float64(Float64(-x) * z));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(N[(1.0 / N[(N[(N[(N[(y$95$m * x + N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(x - N[(N[(N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+298], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t$95$3, -5e+173], t$95$1, If[LessEqual[t$95$3, 2e+83], N[(N[(N[(z * y$95$m), $MachinePrecision] * t$95$2 + x), $MachinePrecision] - N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+304], t$95$1, N[(z * t + N[((-x) * z), $MachinePrecision]), $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))))) < -5.0000000000000003e298

   1. Initial program 70.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -5.0000000000000003e298 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -5.00000000000000034e173 or 2.00000000000000006e83 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 4.9999999999999997e304

   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 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 43.8%

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

   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 86.6%

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

   if -5.00000000000000034e173 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 2.00000000000000006e83

   1. Initial program 98.4%

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

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 98.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 84.5

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

   9. Applied rewrites 84.5%

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

   if 4.9999999999999997e304 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 51.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -5 \cdot 10^{+173}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \tanh \left(\frac{t}{y}\right), x\right) - z \cdot x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \tanh \left(\frac{x}{y\_m}\right)\\ t_2 := \tanh \left(\frac{t}{y\_m}\right)\\ t_3 := x - \left(t\_1 - t\_2\right) \cdot \left(z \cdot y\_m\right)\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y\_m} - t\_1\right) \cdot y\_m, z, x\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y\_m, t\_2, x\right) - z \cdot x\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y\_m, x, \frac{\left(x \cdot x\right) \cdot y\_m}{t}\right)}{t} + y\_m}{t}} \cdot \left(z \cdot y\_m\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (tanh (/ x y_m)))
        (t_2 (tanh (/ t y_m)))
        (t_3 (- x (* (- t_1 t_2) (* z y_m)))))
   (if (<= t_3 -1e+113)
     (fma (* (- (/ t y_m) t_1) y_m) z x)
     (if (<= t_3 2e+83)
       (- (fma (* z y_m) t_2 x) (* z x))
       (if (<= t_3 5e+304)
         (+
          (*
           (/ 1.0 (/ (+ (/ (fma y_m x (/ (* (* x x) y_m) t)) t) y_m) t))
           (* z y_m))
          x)
         (fma z t (* (- x) z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = tanh((x / y_m));
	double t_2 = tanh((t / y_m));
	double t_3 = x - ((t_1 - t_2) * (z * y_m));
	double tmp;
	if (t_3 <= -1e+113) {
		tmp = fma((((t / y_m) - t_1) * y_m), z, x);
	} else if (t_3 <= 2e+83) {
		tmp = fma((z * y_m), t_2, x) - (z * x);
	} else if (t_3 <= 5e+304) {
		tmp = ((1.0 / (((fma(y_m, x, (((x * x) * y_m) / t)) / t) + y_m) / t)) * (z * y_m)) + x;
	} else {
		tmp = fma(z, t, (-x * z));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = tanh(Float64(x / y_m))
	t_2 = tanh(Float64(t / y_m))
	t_3 = Float64(x - Float64(Float64(t_1 - t_2) * Float64(z * y_m)))
	tmp = 0.0
	if (t_3 <= -1e+113)
		tmp = fma(Float64(Float64(Float64(t / y_m) - t_1) * y_m), z, x);
	elseif (t_3 <= 2e+83)
		tmp = Float64(fma(Float64(z * y_m), t_2, x) - Float64(z * x));
	elseif (t_3 <= 5e+304)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y_m, x, Float64(Float64(Float64(x * x) * y_m) / t)) / t) + y_m) / t)) * Float64(z * y_m)) + x);
	else
		tmp = fma(z, t, Float64(Float64(-x) * z));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(x - N[(N[(t$95$1 - t$95$2), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+113], N[(N[(N[(N[(t / y$95$m), $MachinePrecision] - t$95$1), $MachinePrecision] * y$95$m), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t$95$3, 2e+83], N[(N[(N[(z * y$95$m), $MachinePrecision] * t$95$2 + x), $MachinePrecision] - N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+304], N[(N[(N[(1.0 / N[(N[(N[(N[(y$95$m * x + N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * t + N[((-x) * z), $MachinePrecision]), $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))))) < -1e113

   1. Initial program 91.0%

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

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

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

   if -1e113 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 2.00000000000000006e83

   1. Initial program 98.3%

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

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 98.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.5

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

   7. Applied rewrites 86.5%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 86.5

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

   9. Applied rewrites 86.5%

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

   if 2.00000000000000006e83 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 4.9999999999999997e304

   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 40.1

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

   5. Applied rewrites 40.1%

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

   7. Applied rewrites 40.1%

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

   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 84.8%

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

   if 4.9999999999999997e304 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 51.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -1 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right)\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \tanh \left(\frac{t}{y}\right), x\right) - z \cdot x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(\frac{t}{y\_m} - \tanh \left(\frac{x}{y\_m}\right)\right) \cdot y\_m, z, x\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y\_m}\right) - \frac{x}{y\_m}\right) \cdot z, y\_m, x\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y\_m, x, \frac{\left(x \cdot x\right) \cdot y\_m}{t}\right)}{t} + y\_m}{t}} \cdot \left(z \cdot y\_m\right) + x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (fma (* (- (/ t y_m) (tanh (/ x y_m))) y_m) z x)))
   (if (<= x -3.2e+72)
     t_1
     (if (<= x 1.7e+83)
       (fma (* (- (tanh (/ t y_m)) (/ x y_m)) z) y_m x)
       (if (<= x 1.85e+151)
         t_1
         (+
          (*
           (/ 1.0 (/ (+ (/ (fma y_m x (/ (* (* x x) y_m) t)) t) y_m) t))
           (* z y_m))
          x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = fma((((t / y_m) - tanh((x / y_m))) * y_m), z, x);
	double tmp;
	if (x <= -3.2e+72) {
		tmp = t_1;
	} else if (x <= 1.7e+83) {
		tmp = fma(((tanh((t / y_m)) - (x / y_m)) * z), y_m, x);
	} else if (x <= 1.85e+151) {
		tmp = t_1;
	} else {
		tmp = ((1.0 / (((fma(y_m, x, (((x * x) * y_m) / t)) / t) + y_m) / t)) * (z * y_m)) + x;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = fma(Float64(Float64(Float64(t / y_m) - tanh(Float64(x / y_m))) * y_m), z, x)
	tmp = 0.0
	if (x <= -3.2e+72)
		tmp = t_1;
	elseif (x <= 1.7e+83)
		tmp = fma(Float64(Float64(tanh(Float64(t / y_m)) - Float64(x / y_m)) * z), y_m, x);
	elseif (x <= 1.85e+151)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y_m, x, Float64(Float64(Float64(x * x) * y_m) / t)) / t) + y_m) / t)) * Float64(z * y_m)) + x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(t / y$95$m), $MachinePrecision] - N[Tanh[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[x, -3.2e+72], t$95$1, If[LessEqual[x, 1.7e+83], N[(N[(N[(N[Tanh[N[(t / y$95$m), $MachinePrecision]], $MachinePrecision] - N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y$95$m + x), $MachinePrecision], If[LessEqual[x, 1.85e+151], t$95$1, N[(N[(N[(1.0 / N[(N[(N[(N[(y$95$m * x + N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e72 or 1.6999999999999999e83 < x < 1.8499999999999999e151

   1. Initial program 97.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-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. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

   if -3.2000000000000001e72 < x < 1.6999999999999999e83

   1. Initial program 89.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.8

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

   7. Applied rewrites 85.8%

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

   if 1.8499999999999999e151 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\frac{t - x}{y}} \]
   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 45.9

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

   5. Applied rewrites 45.9%

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

   7. Applied rewrites 45.9%

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

   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 94.3%

       \[\leadsto x + \left(y \cdot z\right) \cdot \frac{1}{\frac{-\left(\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y\right)}{\color{blue}{-t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right) \cdot z, y, x\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y\_m, x, \frac{\left(x \cdot x\right) \cdot y\_m}{t}\right)}{t} + y\_m}{t}} \cdot \left(z \cdot y\_m\right) + x\\ \mathbf{elif}\;y\_m \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(t \cdot t, \frac{y\_m}{x}, t \cdot y\_m\right)}{-x} - y\_m}{x}} \cdot \left(z \cdot y\_m\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 2.05e-58)
   (+
    (* (/ 1.0 (/ (+ (/ (fma y_m x (/ (* (* x x) y_m) t)) t) y_m) t)) (* z y_m))
    x)
   (if (<= y_m 2e-9)
     (+
      (*
       (/ 1.0 (/ (- (/ (fma (* t t) (/ y_m x) (* t y_m)) (- x)) y_m) x))
       (* z y_m))
      x)
     (fma (- t x) z x))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 2.05e-58) {
		tmp = ((1.0 / (((fma(y_m, x, (((x * x) * y_m) / t)) / t) + y_m) / t)) * (z * y_m)) + x;
	} else if (y_m <= 2e-9) {
		tmp = ((1.0 / (((fma((t * t), (y_m / x), (t * y_m)) / -x) - y_m) / x)) * (z * y_m)) + x;
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 2.05e-58)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y_m, x, Float64(Float64(Float64(x * x) * y_m) / t)) / t) + y_m) / t)) * Float64(z * y_m)) + x);
	elseif (y_m <= 2e-9)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(Float64(t * t), Float64(y_m / x), Float64(t * y_m)) / Float64(-x)) - y_m) / x)) * Float64(z * y_m)) + x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 2.05e-58], N[(N[(N[(1.0 / N[(N[(N[(N[(y$95$m * x + N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y$95$m, 2e-9], N[(N[(N[(1.0 / N[(N[(N[(N[(N[(t * t), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + N[(t * y$95$m), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] - y$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.05000000000000014e-58

   1. Initial program 96.7%

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

   3. 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 44.2

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

   5. Applied rewrites 44.2%

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

   7. Applied rewrites 44.2%

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

   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 65.3%

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

   if 2.05000000000000014e-58 < y < 2.00000000000000012e-9

   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 44.1

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

   5. Applied rewrites 44.1%

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

   7. Applied rewrites 43.9%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 78.2%

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

   if 2.00000000000000012e-9 < y

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 83.4

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

   5. Applied rewrites 83.4%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(t \cdot t, \frac{y}{x}, t \cdot y\right)}{-x} - y}{x}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y\_m, x, \frac{\left(x \cdot x\right) \cdot y\_m}{t}\right)}{t} + y\_m}{t}} \cdot \left(z \cdot y\_m\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 4.2e-12)
   (+
    (* (/ 1.0 (/ (+ (/ (fma y_m x (/ (* (* x x) y_m) t)) t) y_m) t)) (* z y_m))
    x)
   (fma (- t x) z x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 4.2e-12) {
		tmp = ((1.0 / (((fma(y_m, x, (((x * x) * y_m) / t)) / t) + y_m) / t)) * (z * y_m)) + x;
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 4.2e-12)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y_m, x, Float64(Float64(Float64(x * x) * y_m) / t)) / t) + y_m) / t)) * Float64(z * y_m)) + x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 4.2e-12], N[(N[(N[(1.0 / N[(N[(N[(N[(y$95$m * x + N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.19999999999999988e-12

   1. Initial program 96.9%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   7. Applied rewrites 44.4%

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

   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 64.5%

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

   if 4.19999999999999988e-12 < y

   1. Initial program 82.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 82.3

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

   5. Applied rewrites 82.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{x} \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 7.2e-60) (fma (* (/ t x) x) z x) (fma (- t x) z x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 7.2e-60) {
		tmp = fma(((t / x) * x), z, x);
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 7.2e-60)
		tmp = fma(Float64(Float64(t / x) * x), z, x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 7.2e-60], N[(N[(N[(t / x), $MachinePrecision] * x), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2e-60

   1. Initial program 96.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.5%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 62.4%

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

   if 7.2e-60 < y

   1. Initial program 84.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 77.9

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

   5. Applied rewrites 77.9%

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

Alternative 8: 62.7% accurate, 11.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.052:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (* (- t x) z)))
   (if (<= z -1.55e+25) t_1 (if (<= z 0.052) (fma (- x) z x) t_1))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (t - x) * z;
	double tmp;
	if (z <= -1.55e+25) {
		tmp = t_1;
	} else if (z <= 0.052) {
		tmp = fma(-x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(t - x) * z)
	tmp = 0.0
	if (z <= -1.55e+25)
		tmp = t_1;
	elseif (z <= 0.052)
		tmp = fma(Float64(-x), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.55e+25], t$95$1, If[LessEqual[z, 0.052], N[((-x) * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5499999999999999e25 or 0.0519999999999999976 < z

   1. Initial program 86.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 45.9%

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

   if -1.5499999999999999e25 < z < 0.0519999999999999976

   1. Initial program 98.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 82.5%

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

Alternative 9: 21.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-119}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= t -2.35e-119) (* z t) (if (<= t 4.2e-113) (* (- x) z) (* z t))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (t <= -2.35e-119) {
		tmp = z * t;
	} else if (t <= 4.2e-113) {
		tmp = -x * z;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.35d-119)) then
        tmp = z * t
    else if (t <= 4.2d-113) then
        tmp = -x * z
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if (t <= -2.35e-119) {
		tmp = z * t;
	} else if (t <= 4.2e-113) {
		tmp = -x * z;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if t <= -2.35e-119:
		tmp = z * t
	elif t <= 4.2e-113:
		tmp = -x * z
	else:
		tmp = z * t
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (t <= -2.35e-119)
		tmp = Float64(z * t);
	elseif (t <= 4.2e-113)
		tmp = Float64(Float64(-x) * z);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (t <= -2.35e-119)
		tmp = z * t;
	elseif (t <= 4.2e-113)
		tmp = -x * z;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[t, -2.35e-119], N[(z * t), $MachinePrecision], If[LessEqual[t, 4.2e-113], N[((-x) * z), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.35000000000000001e-119 or 4.2e-113 < t

   1. Initial program 93.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 23.1%

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

   if -2.35000000000000001e-119 < t < 4.2e-113

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 28.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 24.3%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-119}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.1% accurate, 14.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 1.05e-10) (fma (- x) z x) (fma (- t x) z x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 1.05e-10) {
		tmp = fma(-x, z, x);
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 1.05e-10)
		tmp = fma(Float64(-x), z, x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 1.05e-10], N[((-x) * z + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e-10

   1. Initial program 96.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 53.1%

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

   if 1.05e-10 < y

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 83.4

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

   5. Applied rewrites 83.4%

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

Alternative 11: 26.9% accurate, 26.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 62.8

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

5. Applied rewrites 62.8%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 27.1%

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

Alternative 12: 17.4% accurate, 39.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 62.8

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

5. Applied rewrites 62.8%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 18.4%

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

9. Final simplification 18.4%

   \[\leadsto z \cdot t \]
10. Add Preprocessing

Developer Target 1: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024270 
(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))))))