SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.3% → 98.1%

Time: 11.0s

Alternatives: 14

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{x}{y}\right)\\ t_2 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;x - \left(t\_1 - t\_2\right) \cdot \left(z \cdot y\right) \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_2 - t\_1\right) \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ x y))) (t_2 (tanh (/ t y))))
   (if (<= (- x (* (- t_1 t_2) (* z y))) 1e+306)
     (fma (* (- t_2 t_1) z) y x)
     (* (- t x) z))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((x / y));
	double t_2 = tanh((t / y));
	double tmp;
	if ((x - ((t_1 - t_2) * (z * y))) <= 1e+306) {
		tmp = fma(((t_2 - t_1) * z), y, x);
	} else {
		tmp = (t - x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = tanh(Float64(x / y))
	t_2 = tanh(Float64(t / y))
	tmp = 0.0
	if (Float64(x - Float64(Float64(t_1 - t_2) * Float64(z * y))) <= 1e+306)
		tmp = fma(Float64(Float64(t_2 - t_1) * z), y, x);
	else
		tmp = Float64(Float64(t - x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x - N[(N[(t$95$1 - t$95$2), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(N[(t$95$2 - t$95$1), $MachinePrecision] * z), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.00000000000000002e306

   1. Initial program 97.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 1.00000000000000002e306 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 40.0%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   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%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ t_2 := \frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ t_3 := \tanh \left(\frac{x}{y}\right)\\ t_4 := x - \left(t\_3 - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(t, \frac{y}{x} \cdot t, t \cdot y\right)}{x} + y}{-x}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} - t\_3\right) \cdot z, y, x\right)\\ \mathbf{elif}\;t\_4 \leq 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) z))
        (t_2
         (+
          (* (/ 1.0 (/ (+ (/ (fma y x (/ (* (* x x) y) t)) t) y) t)) (* z y))
          x))
        (t_3 (tanh (/ x y)))
        (t_4 (- x (* (- t_3 (tanh (/ t y))) (* z y)))))
   (if (<= t_4 (- INFINITY))
     t_1
     (if (<= t_4 -1e+31)
       t_2
       (if (<= t_4 2e-21)
         (+
          (*
           (/ 1.0 (/ (+ (/ (fma t (* (/ y x) t) (* t y)) x) y) (- x)))
           (* z y))
          x)
         (if (<= t_4 2e+135)
           (fma (* (- (/ t y) t_3) z) y x)
           (if (<= t_4 1e+306) t_2 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * z;
	double t_2 = ((1.0 / (((fma(y, x, (((x * x) * y) / t)) / t) + y) / t)) * (z * y)) + x;
	double t_3 = tanh((x / y));
	double t_4 = x - ((t_3 - tanh((t / y))) * (z * y));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_4 <= -1e+31) {
		tmp = t_2;
	} else if (t_4 <= 2e-21) {
		tmp = ((1.0 / (((fma(t, ((y / x) * t), (t * y)) / x) + y) / -x)) * (z * y)) + x;
	} else if (t_4 <= 2e+135) {
		tmp = fma((((t / y) - t_3) * z), y, x);
	} else if (t_4 <= 1e+306) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * z)
	t_2 = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y, x, Float64(Float64(Float64(x * x) * y) / t)) / t) + y) / t)) * Float64(z * y)) + x)
	t_3 = tanh(Float64(x / y))
	t_4 = Float64(x - Float64(Float64(t_3 - tanh(Float64(t / y))) * Float64(z * y)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_4 <= -1e+31)
		tmp = t_2;
	elseif (t_4 <= 2e-21)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(t, Float64(Float64(y / x) * t), Float64(t * y)) / x) + y) / Float64(-x))) * Float64(z * y)) + x);
	elseif (t_4 <= 2e+135)
		tmp = fma(Float64(Float64(Float64(t / y) - t_3) * z), y, x);
	elseif (t_4 <= 1e+306)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1.0 / N[(N[(N[(N[(y * x + N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$3 = N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(x - N[(N[(t$95$3 - N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$1, If[LessEqual[t$95$4, -1e+31], t$95$2, If[LessEqual[t$95$4, 2e-21], N[(N[(N[(1.0 / N[(N[(N[(N[(t * N[(N[(y / x), $MachinePrecision] * t), $MachinePrecision] + N[(t * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + y), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$4, 2e+135], N[(N[(N[(N[(t / y), $MachinePrecision] - t$95$3), $MachinePrecision] * z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$4, 1e+306], t$95$2, t$95$1]]]]]]]]]

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 or 1.00000000000000002e306 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 51.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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%

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -9.9999999999999996e30 or 1.99999999999999992e135 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.00000000000000002e306

   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 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 38.3%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 61.2%

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

   11. Taylor expanded in t around -inf

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

   13. Applied rewrites 78.0%

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

   if -9.9999999999999996e30 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.99999999999999982e-21

   1. Initial program 98.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\frac{t - x}{y}} \]
   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 59.5

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

   5. Applied rewrites 59.5%

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in x around -inf

      \[\leadsto x + \left(y \cdot z\right) \cdot \frac{1}{-1 \cdot \color{blue}{\frac{y + -1 \cdot \frac{-1 \cdot \frac{{t}^{2} \cdot y}{x} - t \cdot y}{x}}{x}}} \]

   10. Applied rewrites 74.5%

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

   if 1.99999999999999982e-21 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.99999999999999992e135

   1. Initial program 99.8%

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

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

      9. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 72.1

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

   7. Applied rewrites 72.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -\infty:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(t, \frac{y}{x} \cdot t, t \cdot y\right)}{x} + y}{-x}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot z, y, x\right)\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 10^{+306}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ t_2 := \frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ t_3 := \tanh \left(\frac{t}{y}\right)\\ t_4 := x - \left(\tanh \left(\frac{x}{y}\right) - t\_3\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(t\_3 \cdot y, z, \left(\left(-z\right) \cdot y\right) \cdot \frac{x}{y}\right) + x\\ \mathbf{elif}\;t\_4 \leq 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) z))
        (t_2
         (+
          (* (/ 1.0 (/ (+ (/ (fma y x (/ (* (* x x) y) t)) t) y) t)) (* z y))
          x))
        (t_3 (tanh (/ t y)))
        (t_4 (- x (* (- (tanh (/ x y)) t_3) (* z y)))))
   (if (<= t_4 (- INFINITY))
     t_1
     (if (<= t_4 -2e+84)
       t_2
       (if (<= t_4 2e+110)
         (+ (fma (* t_3 y) z (* (* (- z) y) (/ x y))) x)
         (if (<= t_4 1e+306) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * z;
	double t_2 = ((1.0 / (((fma(y, x, (((x * x) * y) / t)) / t) + y) / t)) * (z * y)) + x;
	double t_3 = tanh((t / y));
	double t_4 = x - ((tanh((x / y)) - t_3) * (z * y));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_4 <= -2e+84) {
		tmp = t_2;
	} else if (t_4 <= 2e+110) {
		tmp = fma((t_3 * y), z, ((-z * y) * (x / y))) + x;
	} else if (t_4 <= 1e+306) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * z)
	t_2 = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y, x, Float64(Float64(Float64(x * x) * y) / t)) / t) + y) / t)) * Float64(z * y)) + x)
	t_3 = tanh(Float64(t / y))
	t_4 = Float64(x - Float64(Float64(tanh(Float64(x / y)) - t_3) * Float64(z * y)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_4 <= -2e+84)
		tmp = t_2;
	elseif (t_4 <= 2e+110)
		tmp = Float64(fma(Float64(t_3 * y), z, Float64(Float64(Float64(-z) * y) * Float64(x / y))) + x);
	elseif (t_4 <= 1e+306)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1.0 / N[(N[(N[(N[(y * x + N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$3 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(x - N[(N[(N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$1, If[LessEqual[t$95$4, -2e+84], t$95$2, If[LessEqual[t$95$4, 2e+110], N[(N[(N[(t$95$3 * y), $MachinePrecision] * z + N[(N[((-z) * y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$4, 1e+306], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -inf.0 or 1.00000000000000002e306 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 51.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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%

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -2.00000000000000012e84 or 2e110 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.00000000000000002e306

   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 33.8

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

   5. Applied rewrites 33.8%

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

   7. Applied rewrites 33.8%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 59.0%

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

   11. Taylor expanded in t around -inf

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

   13. Applied rewrites 77.6%

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

   if -2.00000000000000012e84 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 2e110

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-neg.f64 83.8

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

   7. Applied rewrites 83.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -\infty:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -2 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right) \cdot y, z, \left(\left(-z\right) \cdot y\right) \cdot \frac{x}{y}\right) + x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 10^{+306}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ t_2 := \frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ t_3 := \tanh \left(\frac{t}{y}\right)\\ t_4 := x - \left(\tanh \left(\frac{x}{y}\right) - t\_3\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(t\_3 - \frac{x}{y}, z \cdot y, x\right)\\ \mathbf{elif}\;t\_4 \leq 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) z))
        (t_2
         (+
          (* (/ 1.0 (/ (+ (/ (fma y x (/ (* (* x x) y) t)) t) y) t)) (* z y))
          x))
        (t_3 (tanh (/ t y)))
        (t_4 (- x (* (- (tanh (/ x y)) t_3) (* z y)))))
   (if (<= t_4 (- INFINITY))
     t_1
     (if (<= t_4 -2e+84)
       t_2
       (if (<= t_4 2e+110)
         (fma (- t_3 (/ x y)) (* z y) x)
         (if (<= t_4 1e+306) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * z;
	double t_2 = ((1.0 / (((fma(y, x, (((x * x) * y) / t)) / t) + y) / t)) * (z * y)) + x;
	double t_3 = tanh((t / y));
	double t_4 = x - ((tanh((x / y)) - t_3) * (z * y));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_4 <= -2e+84) {
		tmp = t_2;
	} else if (t_4 <= 2e+110) {
		tmp = fma((t_3 - (x / y)), (z * y), x);
	} else if (t_4 <= 1e+306) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * z)
	t_2 = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y, x, Float64(Float64(Float64(x * x) * y) / t)) / t) + y) / t)) * Float64(z * y)) + x)
	t_3 = tanh(Float64(t / y))
	t_4 = Float64(x - Float64(Float64(tanh(Float64(x / y)) - t_3) * Float64(z * y)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_4 <= -2e+84)
		tmp = t_2;
	elseif (t_4 <= 2e+110)
		tmp = fma(Float64(t_3 - Float64(x / y)), Float64(z * y), x);
	elseif (t_4 <= 1e+306)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1.0 / N[(N[(N[(N[(y * x + N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$3 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(x - N[(N[(N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$1, If[LessEqual[t$95$4, -2e+84], t$95$2, If[LessEqual[t$95$4, 2e+110], N[(N[(t$95$3 - N[(x / y), $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$4, 1e+306], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -inf.0 or 1.00000000000000002e306 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 51.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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%

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -2.00000000000000012e84 or 2e110 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.00000000000000002e306

   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 33.8

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

   5. Applied rewrites 33.8%

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

   7. Applied rewrites 33.8%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 59.0%

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

   11. Taylor expanded in t around -inf

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

   13. Applied rewrites 77.6%

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

   if -2.00000000000000012e84 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 2e110

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 83.7

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

   7. Applied rewrites 83.7%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -\infty:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -2 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}, z \cdot y, x\right)\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 10^{+306}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ t_2 := x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right)\\ t_3 := \frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(t, \frac{y}{x} \cdot t, t \cdot y\right)}{x} + y}{-x}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;t\_2 \leq 10^{+306}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) z))
        (t_2 (- x (* (- (tanh (/ x y)) (tanh (/ t y))) (* z y))))
        (t_3
         (+
          (* (/ 1.0 (/ (+ (/ (fma y x (/ (* (* x x) y) t)) t) y) t)) (* z y))
          x)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e+31)
       t_3
       (if (<= t_2 2e-21)
         (+
          (*
           (/ 1.0 (/ (+ (/ (fma t (* (/ y x) t) (* t y)) x) y) (- x)))
           (* z y))
          x)
         (if (<= t_2 1e+306) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * z;
	double t_2 = x - ((tanh((x / y)) - tanh((t / y))) * (z * y));
	double t_3 = ((1.0 / (((fma(y, x, (((x * x) * y) / t)) / t) + y) / t)) * (z * y)) + x;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e+31) {
		tmp = t_3;
	} else if (t_2 <= 2e-21) {
		tmp = ((1.0 / (((fma(t, ((y / x) * t), (t * y)) / x) + y) / -x)) * (z * y)) + x;
	} else if (t_2 <= 1e+306) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * z)
	t_2 = Float64(x - Float64(Float64(tanh(Float64(x / y)) - tanh(Float64(t / y))) * Float64(z * y)))
	t_3 = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y, x, Float64(Float64(Float64(x * x) * y) / t)) / t) + y) / t)) * Float64(z * y)) + x)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e+31)
		tmp = t_3;
	elseif (t_2 <= 2e-21)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(t, Float64(Float64(y / x) * t), Float64(t * y)) / x) + y) / Float64(-x))) * Float64(z * y)) + x);
	elseif (t_2 <= 1e+306)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1.0 / N[(N[(N[(N[(y * x + N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e+31], t$95$3, If[LessEqual[t$95$2, 2e-21], N[(N[(N[(1.0 / N[(N[(N[(N[(t * N[(N[(y / x), $MachinePrecision] * t), $MachinePrecision] + N[(t * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + y), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 1e+306], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -inf.0 or 1.00000000000000002e306 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 51.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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%

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -9.9999999999999996e30 or 1.99999999999999982e-21 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.00000000000000002e306

   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 43.9

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

   5. Applied rewrites 43.9%

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

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

   10. Applied rewrites 61.0%

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

   11. Taylor expanded in t around -inf

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

   13. Applied rewrites 73.0%

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

   if -9.9999999999999996e30 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.99999999999999982e-21

   1. Initial program 98.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\frac{t - x}{y}} \]
   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 59.5

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

   5. Applied rewrites 59.5%

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in x around -inf

      \[\leadsto x + \left(y \cdot z\right) \cdot \frac{1}{-1 \cdot \color{blue}{\frac{y + -1 \cdot \frac{-1 \cdot \frac{{t}^{2} \cdot y}{x} - t \cdot y}{x}}{x}}} \]

   10. Applied rewrites 74.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -\infty:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(t, \frac{y}{x} \cdot t, t \cdot y\right)}{x} + y}{-x}} \cdot \left(z \cdot y\right) + x\\ \mathbf{elif}\;x - \left(\tanh \left(\frac{x}{y}\right) - \tanh \left(\frac{t}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 10^{+306}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5e+52)
   (+ (* (/ 1.0 (/ (+ (/ (fma y x (/ (* (* x x) y) t)) t) y) t)) (* z y)) x)
   (fma (- t x) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5e+52) {
		tmp = ((1.0 / (((fma(y, x, (((x * x) * y) / t)) / t) + y) / t)) * (z * y)) + x;
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5e+52)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(fma(y, x, Float64(Float64(Float64(x * x) * y) / t)) / t) + y) / t)) * Float64(z * y)) + x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5e+52], N[(N[(N[(1.0 / N[(N[(N[(N[(y * x + N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e52

   1. Initial program 96.6%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 45.2

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

   5. Applied rewrites 45.2%

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

   7. Applied rewrites 45.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 54.0%

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

   11. Taylor expanded in t around -inf

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

   13. Applied rewrites 60.1%

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

   if 5e52 < y

   1. Initial program 81.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 87.4

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

   5. Applied rewrites 87.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(y, x, \frac{\left(x \cdot x\right) \cdot y}{t}\right)}{t} + y}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{-216}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{elif}\;y \leq 23:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, x, y\right)}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.32e-216)
   (fma (- x) z x)
   (if (<= y 23.0)
     (+ (* (/ 1.0 (/ (fma (/ y t) x y) t)) (* z y)) x)
     (fma (- t x) z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.32e-216) {
		tmp = fma(-x, z, x);
	} else if (y <= 23.0) {
		tmp = ((1.0 / (fma((y / t), x, y) / t)) * (z * y)) + x;
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.32e-216)
		tmp = fma(Float64(-x), z, x);
	elseif (y <= 23.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(fma(Float64(y / t), x, y) / t)) * Float64(z * y)) + x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.32e-216], N[((-x) * z + x), $MachinePrecision], If[LessEqual[y, 23.0], N[(N[(N[(1.0 / N[(N[(N[(y / t), $MachinePrecision] * x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.31999999999999997e-216

   1. Initial program 95.1%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 59.9

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

   5. Applied rewrites 59.9%

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

   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 52.9%

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

   if 1.31999999999999997e-216 < y < 23

   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 35.4

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

   5. Applied rewrites 35.4%

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 55.9%

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

   if 23 < y

   1. Initial program 83.7%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{-216}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{elif}\;y \leq 23:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, x, y\right)}{t}} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{elif}\;y \leq 3300000:\\ \;\;\;\;\left(\frac{\frac{y}{x} \cdot t}{y \cdot y} \cdot x\right) \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.2e-105)
   (fma (- x) z x)
   (if (<= y 3300000.0)
     (+ (* (* (/ (* (/ y x) t) (* y y)) x) (* z y)) x)
     (fma (- t x) z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.2e-105) {
		tmp = fma(-x, z, x);
	} else if (y <= 3300000.0) {
		tmp = (((((y / x) * t) / (y * y)) * x) * (z * y)) + x;
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.2e-105)
		tmp = fma(Float64(-x), z, x);
	elseif (y <= 3300000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(y / x) * t) / Float64(y * y)) * x) * Float64(z * y)) + x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.2e-105], N[((-x) * z + x), $MachinePrecision], If[LessEqual[y, 3300000.0], N[(N[(N[(N[(N[(N[(y / x), $MachinePrecision] * t), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.19999999999999981e-105

   1. Initial program 96.0%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 54.9

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

   5. Applied rewrites 54.9%

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

   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 51.9%

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

   if 3.19999999999999981e-105 < y < 3.3e6

   1. Initial program 99.8%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 59.9%

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

   10. Applied rewrites 63.3%

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

   if 3.3e6 < y

   1. Initial program 83.1%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 83.6

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

   5. Applied rewrites 83.6%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{elif}\;y \leq 3300000:\\ \;\;\;\;\left(\frac{\frac{y}{x} \cdot t}{y \cdot y} \cdot x\right) \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{elif}\;y \leq 3300000:\\ \;\;\;\;\frac{t}{y} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.75e-202)
   (fma (- x) z x)
   (if (<= y 3300000.0) (+ (* (/ t y) (* z y)) x) (fma (- t x) z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.75e-202) {
		tmp = fma(-x, z, x);
	} else if (y <= 3300000.0) {
		tmp = ((t / y) * (z * y)) + x;
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.75e-202)
		tmp = fma(Float64(-x), z, x);
	elseif (y <= 3300000.0)
		tmp = Float64(Float64(Float64(t / y) * Float64(z * y)) + x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.75e-202], N[((-x) * z + x), $MachinePrecision], If[LessEqual[y, 3300000.0], N[(N[(N[(t / y), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.75e-202

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-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 58.7

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

   5. Applied rewrites 58.7%

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

   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 52.5%

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

   if 1.75e-202 < y < 3.3e6

   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 36.2

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

   5. Applied rewrites 36.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 55.5%

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

   if 3.3e6 < y

   1. Initial program 83.1%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 83.6

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

   5. Applied rewrites 83.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{elif}\;y \leq 3300000:\\ \;\;\;\;\frac{t}{y} \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ \mathbf{if}\;z \leq -0.66:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2050000:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) z)))
   (if (<= z -0.66) t_1 (if (<= z 2050000.0) (fma (- x) z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * z;
	double tmp;
	if (z <= -0.66) {
		tmp = t_1;
	} else if (z <= 2050000.0) {
		tmp = fma(-x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * z)
	tmp = 0.0
	if (z <= -0.66)
		tmp = t_1;
	elseif (z <= 2050000.0)
		tmp = fma(Float64(-x), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -0.66], t$95$1, If[LessEqual[z, 2050000.0], N[((-x) * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.660000000000000031 or 2.05e6 < z

   1. Initial program 86.7%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 44.1%

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

   if -0.660000000000000031 < z < 2.05e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 75.3

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

   5. Applied rewrites 75.3%

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

   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 80.9%

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

Alternative 11: 20.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-160}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.7e-160) (* t z) (if (<= t 1.6e-8) (* (- x) z) (* t z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.7e-160) {
		tmp = t * z;
	} else if (t <= 1.6e-8) {
		tmp = -x * z;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.7d-160)) then
        tmp = t * z
    else if (t <= 1.6d-8) then
        tmp = -x * z
    else
        tmp = t * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.7e-160) {
		tmp = t * z;
	} else if (t <= 1.6e-8) {
		tmp = -x * z;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.7e-160:
		tmp = t * z
	elif t <= 1.6e-8:
		tmp = -x * z
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.7e-160)
		tmp = Float64(t * z);
	elseif (t <= 1.6e-8)
		tmp = Float64(Float64(-x) * z);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.7e-160)
		tmp = t * z;
	elseif (t <= 1.6e-8)
		tmp = -x * z;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.7e-160], N[(t * z), $MachinePrecision], If[LessEqual[t, 1.6e-8], N[((-x) * z), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.70000000000000011e-160 or 1.6000000000000001e-8 < t

   1. Initial program 96.3%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 24.5%

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

   if -1.70000000000000011e-160 < t < 1.6000000000000001e-8

   1. Initial program 88.9%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 29.9%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 23.6%

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

Alternative 12: 59.6% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2e-91) (fma (- x) z x) (fma (- t x) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2e-91) {
		tmp = fma(-x, z, x);
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2e-91)
		tmp = fma(Float64(-x), z, x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2e-91], N[((-x) * z + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000004e-91

   1. Initial program 96.1%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 53.5

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

   5. Applied rewrites 53.5%

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

   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 51.2%

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

   if 2.00000000000000004e-91 < y

   1. Initial program 87.6%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

Alternative 13: 27.2% accurate, 26.6× speedup

Math

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

FPCore

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

C

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

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 = (t - x) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 60.1

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 27.9%

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

Alternative 14: 17.1% accurate, 39.8× speedup

Math

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

FPCore

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

C

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

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 = t * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 60.1

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 18.8%

   \[\leadsto t \cdot \color{blue}{z} \]
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 2024244 
(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))))))