SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.5% → 97.0%

Time: 13.8s

Alternatives: 15

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: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-tanh.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tanh.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 97.9

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

4. Applied egg-rr 97.9%

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

   1. lift-/.f64 N/A

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

   2. lift-tanh.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-tanh.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 97.9

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

6. Applied egg-rr 97.9%

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

7. Final simplification 97.9%

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

Alternative 2: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-tanh.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tanh.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 97.9

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

4. Applied egg-rr 97.9%

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

Alternative 3: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{-1}{\frac{-1 - \frac{x + \frac{x \cdot x}{t}}{t}}{t}}, x\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right), z, \frac{z \cdot \left(-x\right)}{y}\right), y, x\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ -1.0 (/ (- -1.0 (/ (+ x (/ (* x x) t)) t)) t)) x)))
   (if (<= x -1.8e+23)
     t_1
     (if (<= x 7.6e-11)
       (fma (fma (tanh (/ t y)) z (/ (* z (- x)) y)) y x)
       (if (<= x 4e+197) (fma (* z (- (/ t y) (tanh (/ x y)))) y x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (-1.0 / ((-1.0 - ((x + ((x * x) / t)) / t)) / t)), x);
	double tmp;
	if (x <= -1.8e+23) {
		tmp = t_1;
	} else if (x <= 7.6e-11) {
		tmp = fma(fma(tanh((t / y)), z, ((z * -x) / y)), y, x);
	} else if (x <= 4e+197) {
		tmp = fma((z * ((t / y) - tanh((x / y)))), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(-1.0 / Float64(Float64(-1.0 - Float64(Float64(x + Float64(Float64(x * x) / t)) / t)) / t)), x)
	tmp = 0.0
	if (x <= -1.8e+23)
		tmp = t_1;
	elseif (x <= 7.6e-11)
		tmp = fma(fma(tanh(Float64(t / y)), z, Float64(Float64(z * Float64(-x)) / y)), y, x);
	elseif (x <= 4e+197)
		tmp = fma(Float64(z * Float64(Float64(t / y) - tanh(Float64(x / y)))), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(-1.0 / N[(N[(-1.0 - N[(N[(x + N[(N[(x * x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -1.8e+23], t$95$1, If[LessEqual[x, 7.6e-11], N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * z + N[(N[(z * (-x)), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[x, 4e+197], N[(N[(z * N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e23 or 3.9999999999999998e197 < x

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 64.3

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

   5. Simplified 64.3%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 27.2

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

   7. Applied egg-rr 27.2%

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

   8. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{{x}^{2}}{t} - -1 \cdot x}{t} - 1}{t}\right)}}, x\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{\color{blue}{\frac{-1 \cdot \frac{\frac{{x}^{2}}{t} - -1 \cdot x}{t} - 1}{\mathsf{neg}\left(t\right)}}}, x\right) \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

   10. Simplified 82.0%

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

   if -1.7999999999999999e23 < x < 7.5999999999999996e-11

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tanh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tanh.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 96.0

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

   4. Applied egg-rr 96.0%

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

      1. lift-/.f64 N/A

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

      2. lift-tanh.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tanh.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 96.0

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 84.5

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

   9. Simplified 84.5%

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

   if 7.5999999999999996e-11 < x < 3.9999999999999998e197

   1. Initial program 94.5%

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

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

      3. lift-tanh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tanh.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 85.7

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

   7. Simplified 85.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-1}{\frac{-1 - \frac{x + \frac{x \cdot x}{t}}{t}}{t}}, x\right)\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right), z, \frac{z \cdot \left(-x\right)}{y}\right), y, x\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-1}{\frac{-1 - \frac{x + \frac{x \cdot x}{t}}{t}}{t}}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{-1}{\frac{-1 - \frac{x + \frac{x \cdot x}{t}}{t}}{t}}, x\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), y, x\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ -1.0 (/ (- -1.0 (/ (+ x (/ (* x x) t)) t)) t)) x)))
   (if (<= x -1.8e+23)
     t_1
     (if (<= x 5e-13)
       (fma (* z (- (tanh (/ t y)) (/ x y))) y x)
       (if (<= x 4e+197) (fma (* z (- (/ t y) (tanh (/ x y)))) y x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (-1.0 / ((-1.0 - ((x + ((x * x) / t)) / t)) / t)), x);
	double tmp;
	if (x <= -1.8e+23) {
		tmp = t_1;
	} else if (x <= 5e-13) {
		tmp = fma((z * (tanh((t / y)) - (x / y))), y, x);
	} else if (x <= 4e+197) {
		tmp = fma((z * ((t / y) - tanh((x / y)))), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(-1.0 / Float64(Float64(-1.0 - Float64(Float64(x + Float64(Float64(x * x) / t)) / t)) / t)), x)
	tmp = 0.0
	if (x <= -1.8e+23)
		tmp = t_1;
	elseif (x <= 5e-13)
		tmp = fma(Float64(z * Float64(tanh(Float64(t / y)) - Float64(x / y))), y, x);
	elseif (x <= 4e+197)
		tmp = fma(Float64(z * Float64(Float64(t / y) - tanh(Float64(x / y)))), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(-1.0 / N[(N[(-1.0 - N[(N[(x + N[(N[(x * x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -1.8e+23], t$95$1, If[LessEqual[x, 5e-13], N[(N[(z * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[x, 4e+197], N[(N[(z * N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e23 or 3.9999999999999998e197 < x

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 64.3

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

   5. Simplified 64.3%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 27.2

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

   7. Applied egg-rr 27.2%

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

   8. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{{x}^{2}}{t} - -1 \cdot x}{t} - 1}{t}\right)}}, x\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{\color{blue}{\frac{-1 \cdot \frac{\frac{{x}^{2}}{t} - -1 \cdot x}{t} - 1}{\mathsf{neg}\left(t\right)}}}, x\right) \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

   10. Simplified 82.0%

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

   if -1.7999999999999999e23 < x < 4.9999999999999999e-13

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 80.2

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

   5. Simplified 80.2%

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

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

      3. lift-tanh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 83.8

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

   7. Applied egg-rr 83.8%

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

   if 4.9999999999999999e-13 < x < 3.9999999999999998e197

   1. Initial program 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tanh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tanh.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 85.3

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

   7. Simplified 85.3%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-1}{\frac{-1 - \frac{x + \frac{x \cdot x}{t}}{t}}{t}}, x\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), y, x\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-1}{\frac{-1 - \frac{x + \frac{x \cdot x}{t}}{t}}{t}}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{-1}{\frac{-1 - \frac{x + \frac{x \cdot x}{t}}{t}}{t}}, x\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ -1.0 (/ (- -1.0 (/ (+ x (/ (* x x) t)) t)) t)) x)))
   (if (<= x -1.8e+23)
     t_1
     (if (<= x 1.32e+163) (fma (* z (- (tanh (/ t y)) (/ x y))) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (-1.0 / ((-1.0 - ((x + ((x * x) / t)) / t)) / t)), x);
	double tmp;
	if (x <= -1.8e+23) {
		tmp = t_1;
	} else if (x <= 1.32e+163) {
		tmp = fma((z * (tanh((t / y)) - (x / y))), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(-1.0 / Float64(Float64(-1.0 - Float64(Float64(x + Float64(Float64(x * x) / t)) / t)) / t)), x)
	tmp = 0.0
	if (x <= -1.8e+23)
		tmp = t_1;
	elseif (x <= 1.32e+163)
		tmp = fma(Float64(z * Float64(tanh(Float64(t / y)) - Float64(x / y))), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(-1.0 / N[(N[(-1.0 - N[(N[(x + N[(N[(x * x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -1.8e+23], t$95$1, If[LessEqual[x, 1.32e+163], N[(N[(z * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7999999999999999e23 or 1.31999999999999995e163 < x

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 63.2

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

   5. Simplified 63.2%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 26.2

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

   7. Applied egg-rr 26.2%

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

   8. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{{x}^{2}}{t} - -1 \cdot x}{t} - 1}{t}\right)}}, x\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{\color{blue}{\frac{-1 \cdot \frac{\frac{{x}^{2}}{t} - -1 \cdot x}{t} - 1}{\mathsf{neg}\left(t\right)}}}, x\right) \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

   10. Simplified 80.4%

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

   if -1.7999999999999999e23 < x < 1.31999999999999995e163

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 78.1

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

   5. Simplified 78.1%

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

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

      3. lift-tanh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 82.8

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

   7. Applied egg-rr 82.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-1}{\frac{-1 - \frac{x + \frac{x \cdot x}{t}}{t}}{t}}, x\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-1}{\frac{-1 - \frac{x + \frac{x \cdot x}{t}}{t}}{t}}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-1}{\frac{1 + \frac{t + \frac{t \cdot t}{x}}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.8e-39)
   (fma z (/ -1.0 (/ (+ 1.0 (/ (+ t (/ (* t t) x)) x)) x)) x)
   (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.8e-39) {
		tmp = fma(z, (-1.0 / ((1.0 + ((t + ((t * t) / x)) / x)) / x)), x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.8e-39)
		tmp = fma(z, Float64(-1.0 / Float64(Float64(1.0 + Float64(Float64(t + Float64(Float64(t * t) / x)) / x)) / x)), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.8e-39], N[(z * N[(-1.0 / N[(N[(1.0 + N[(N[(t + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000031e-39

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 54.6

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

   5. Simplified 54.6%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 40.6

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

   7. Applied egg-rr 40.6%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

   10. Simplified 64.2%

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

   if 4.80000000000000031e-39 < y

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 86.4

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

   5. Simplified 86.4%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-1}{\frac{1 + \frac{t + \frac{t \cdot t}{x}}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.8e-39)
   (fma z (/ 1.0 (/ (- -1.0 (/ t x)) x)) x)
   (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.8e-39) {
		tmp = fma(z, (1.0 / ((-1.0 - (t / x)) / x)), x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.8e-39)
		tmp = fma(z, Float64(1.0 / Float64(Float64(-1.0 - Float64(t / x)) / x)), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.8e-39], N[(z * N[(1.0 / N[(N[(-1.0 - N[(t / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000031e-39

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 54.6

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

   5. Simplified 54.6%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 40.6

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

   7. Applied egg-rr 40.6%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-frac N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 61.0

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

   10. Simplified 61.0%

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

   if 4.80000000000000031e-39 < y

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 86.4

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

   5. Simplified 86.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{-1}, \frac{y \cdot x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.8e-39) (fma (/ z -1.0) (/ (* y x) y) x) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.8e-39) {
		tmp = fma((z / -1.0), ((y * x) / y), x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.8e-39)
		tmp = fma(Float64(z / -1.0), Float64(Float64(y * x) / y), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.8e-39], N[(N[(z / -1.0), $MachinePrecision] * N[(N[(y * x), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000031e-39

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 64.9

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

   5. Simplified 64.9%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 48.4

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

   8. Simplified 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. times-frac N/A

          \[\leadsto \color{blue}{\frac{z}{-1} \cdot \frac{y \cdot x}{y}} + x \]

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 52.5

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

   10. Applied egg-rr 52.5%

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

   if 4.80000000000000031e-39 < y

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 86.4

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

   5. Simplified 86.4%

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

Alternative 9: 56.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-39}:\\ \;\;\;\;x - \frac{z \cdot \left(y \cdot x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.5e-39) (- x (/ (* z (* y x)) y)) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.5e-39) {
		tmp = x - ((z * (y * x)) / y);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.5e-39)
		tmp = Float64(x - Float64(Float64(z * Float64(y * x)) / y));
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.5e-39], N[(x - N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5e-39

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 64.9

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

   5. Simplified 64.9%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 48.4

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

   8. Simplified 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 48.4

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 54.2

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

   10. Applied egg-rr 54.2%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 52.0

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

   12. Applied egg-rr 52.0%

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

   if 3.5e-39 < y

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 86.4

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

   5. Simplified 86.4%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-39}:\\ \;\;\;\;x - \frac{z \cdot \left(y \cdot x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-39}:\\ \;\;\;\;x - \frac{y \cdot \left(z \cdot x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.5e-39) (- x (/ (* y (* z x)) y)) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.5e-39) {
		tmp = x - ((y * (z * x)) / y);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.5e-39)
		tmp = Float64(x - Float64(Float64(y * Float64(z * x)) / y));
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.5e-39], N[(x - N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5e-39

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 64.9

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

   5. Simplified 64.9%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 48.4

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

   8. Simplified 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 48.4

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 54.2

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

   10. Applied egg-rr 54.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-/.f64 N/A

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

       7. lift-neg.f64 N/A

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

       8. distribute-frac-neg2 N/A

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

       9. unsub-neg N/A

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

       10. lower--.f64 N/A

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

       11. lower-/.f64 54.2

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

   12. Applied egg-rr 54.2%

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

   if 3.5e-39 < y

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 86.4

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

   5. Simplified 86.4%

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

Alternative 11: 61.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(t - x\right)\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4500000000:\\ \;\;\;\;\mathsf{fma}\left(z, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- t x))))
   (if (<= z -1.8e-19) t_1 (if (<= z 4500000000.0) (fma z (- x) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (t - x);
	double tmp;
	if (z <= -1.8e-19) {
		tmp = t_1;
	} else if (z <= 4500000000.0) {
		tmp = fma(z, -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(t - x))
	tmp = 0.0
	if (z <= -1.8e-19)
		tmp = t_1;
	elseif (z <= 4500000000.0)
		tmp = fma(z, Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e-19], t$95$1, If[LessEqual[z, 4500000000.0], N[(z * (-x) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8000000000000001e-19 or 4.5e9 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 46.2

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

   5. Simplified 46.2%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 45.9

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

   8. Simplified 45.9%

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

   if -1.8000000000000001e-19 < z < 4.5e9

   1. Initial program 98.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.9

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

   8. Simplified 87.9%

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

Alternative 12: 19.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-70}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.5) (* t z) (if (<= t 6e-70) (* z (- x)) (* t z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.5) {
		tmp = t * z;
	} else if (t <= 6e-70) {
		tmp = z * -x;
	} 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 <= (-4.5d0)) then
        tmp = t * z
    else if (t <= 6d-70) then
        tmp = z * -x
    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 <= -4.5) {
		tmp = t * z;
	} else if (t <= 6e-70) {
		tmp = z * -x;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.5:
		tmp = t * z
	elif t <= 6e-70:
		tmp = z * -x
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.5)
		tmp = Float64(t * z);
	elseif (t <= 6e-70)
		tmp = Float64(z * Float64(-x));
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -4.5)
		tmp = t * z;
	elseif (t <= 6e-70)
		tmp = z * -x;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.5], N[(t * z), $MachinePrecision], If[LessEqual[t, 6e-70], N[(z * (-x)), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.5 or 6.0000000000000003e-70 < t

   1. Initial program 94.4%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 53.8

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

   5. Simplified 53.8%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.9

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

   8. Simplified 24.9%

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

   if -4.5 < t < 6.0000000000000003e-70

   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 x around 0

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

      1. lower-/.f64 70.3

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

   5. Simplified 70.3%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 65.0

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

   8. Simplified 65.0%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot z\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{z \cdot x}\right) \]

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

          \[\leadsto \color{blue}{z \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

       4. mul-1-neg N/A

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

       5. lower-*.f64 N/A

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

       6. mul-1-neg N/A

          \[\leadsto z \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

       7. lower-neg.f64 23.1

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

   11. Simplified 23.1%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-70}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(z, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.8e-39) (fma z (- x) x) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.8e-39) {
		tmp = fma(z, -x, x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.8e-39)
		tmp = fma(z, Float64(-x), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.8e-39], N[(z * (-x) + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000031e-39

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 54.6

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

   5. Simplified 54.6%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.5

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

   8. Simplified 55.5%

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

   if 4.80000000000000031e-39 < y

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 86.4

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

   5. Simplified 86.4%

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

Alternative 14: 25.9% accurate, 26.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower--.f64 63.7

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

5. Simplified 63.7%

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

6. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 27.6

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

8. Simplified 27.6%

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

Alternative 15: 16.6% 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 94.7%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower--.f64 63.7

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

5. Simplified 63.7%

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

6. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 18.1

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

8. Simplified 18.1%

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

9. Final simplification 18.1%

   \[\leadsto t \cdot z \]
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 2024208 
(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))))))