SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.9% → 98.0%

Time: 11.8s

Alternatives: 11

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: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 98.5

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

6. Applied rewrites 98.5%

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

7. Final simplification 98.5%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 98.5

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

4. Applied rewrites 98.5%

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

Alternative 3: 80.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\frac{t}{y}, y, \left(-y\right) \cdot \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right), y, -x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (fma (/ t y) y (* (- y) (tanh (/ x y)))) z x)))
   (if (<= x -3.8e+86)
     t_1
     (if (<= x 2.2e+136) (fma (fma (tanh (/ t y)) y (- x)) z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(fma((t / y), y, (-y * tanh((x / y)))), z, x);
	double tmp;
	if (x <= -3.8e+86) {
		tmp = t_1;
	} else if (x <= 2.2e+136) {
		tmp = fma(fma(tanh((t / y)), y, -x), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(fma(Float64(t / y), y, Float64(Float64(-y) * tanh(Float64(x / y)))), z, x)
	tmp = 0.0
	if (x <= -3.8e+86)
		tmp = t_1;
	elseif (x <= 2.2e+136)
		tmp = fma(fma(tanh(Float64(t / y)), y, Float64(-x)), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t / y), $MachinePrecision] * y + N[((-y) * N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[x, -3.8e+86], t$95$1, If[LessEqual[x, 2.2e+136], N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * y + (-x)), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.79999999999999978e86 or 2.1999999999999999e136 < x

   1. Initial program 98.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 83.8

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

   9. Applied rewrites 83.8%

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

   if -3.79999999999999978e86 < x < 2.1999999999999999e136

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 97.7

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.6

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

   9. Applied rewrites 85.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{t}{y}, y, \left(-y\right) \cdot \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right), y, -x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{t}{y}, y, \left(-y\right) \cdot \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right), z \cdot y, x\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right), y, -x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- (/ t y) (tanh (/ x y))) (* z y) x)))
   (if (<= x -3.8e+86)
     t_1
     (if (<= x 9e+136) (fma (fma (tanh (/ t y)) y (- x)) z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(((t / y) - tanh((x / y))), (z * y), x);
	double tmp;
	if (x <= -3.8e+86) {
		tmp = t_1;
	} else if (x <= 9e+136) {
		tmp = fma(fma(tanh((t / y)), y, -x), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(Float64(t / y) - tanh(Float64(x / y))), Float64(z * y), x)
	tmp = 0.0
	if (x <= -3.8e+86)
		tmp = t_1;
	elseif (x <= 9e+136)
		tmp = fma(fma(tanh(Float64(t / y)), y, Float64(-x)), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -3.8e+86], t$95$1, If[LessEqual[x, 9e+136], N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * y + (-x)), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.79999999999999978e86 or 8.9999999999999999e136 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.7

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

   7. Applied rewrites 81.7%

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

   if -3.79999999999999978e86 < x < 8.9999999999999999e136

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 97.7

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.6

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

   9. Applied rewrites 85.6%

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

Alternative 5: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \frac{t}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right), y, -x\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.8e+244)
   (fma (* z y) (/ t y) x)
   (fma (fma (tanh (/ t y)) y (- x)) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.8e+244) {
		tmp = fma((z * y), (t / y), x);
	} else {
		tmp = fma(fma(tanh((t / y)), y, -x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.8e+244)
		tmp = fma(Float64(z * y), Float64(t / y), x);
	else
		tmp = fma(fma(tanh(Float64(t / y)), y, Float64(-x)), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.8e+244], N[(N[(z * y), $MachinePrecision] * N[(t / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * y + (-x)), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8000000000000003e244

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-out-- N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-out-- N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 11.9

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

   7. Applied rewrites 11.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 11.4

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

   9. Applied rewrites 11.4%

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

   10. Taylor expanded in t around inf

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

   12. Applied rewrites 92.9%

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

   if -5.8000000000000003e244 < x

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 98.4

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

   6. Applied rewrites 98.4%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.3

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

   9. Applied rewrites 78.3%

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

Alternative 6: 58.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \frac{t}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 9.5e+34) (fma (* z y) (/ t y) x) (fma (- t x) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e+34) {
		tmp = fma((z * y), (t / y), x);
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.5e+34)
		tmp = fma(Float64(z * y), Float64(t / y), x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9.5e+34], N[(N[(z * y), $MachinePrecision] * N[(t / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.4999999999999999e34

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in y around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-out-- N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-out-- N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 41.1

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

   7. Applied rewrites 41.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 37.4

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

   9. Applied rewrites 37.4%

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

   10. Taylor expanded in t around inf

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

   12. Applied rewrites 52.7%

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

   if 9.4999999999999999e34 < y

   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. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 82.5

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

   5. Applied rewrites 82.5%

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

Alternative 7: 63.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.053:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) z)))
   (if (<= z -5.6e-6) t_1 (if (<= z 0.053) (fma (- x) z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * z;
	double tmp;
	if (z <= -5.6e-6) {
		tmp = t_1;
	} else if (z <= 0.053) {
		tmp = fma(-x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * z)
	tmp = 0.0
	if (z <= -5.6e-6)
		tmp = t_1;
	elseif (z <= 0.053)
		tmp = fma(Float64(-x), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -5.6e-6], t$95$1, If[LessEqual[z, 0.053], N[((-x) * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.59999999999999975e-6 or 0.0529999999999999985 < z

   1. Initial program 88.8%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 39.8%

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

   if -5.59999999999999975e-6 < z < 0.0529999999999999985

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 86.4%

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

Alternative 8: 21.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-135}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e-135) (* z t) (if (<= t 3.8e-65) (* (- x) z) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e-135) {
		tmp = z * t;
	} else if (t <= 3.8e-65) {
		tmp = -x * z;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9d-135)) then
        tmp = z * t
    else if (t <= 3.8d-65) then
        tmp = -x * z
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e-135) {
		tmp = z * t;
	} else if (t <= 3.8e-65) {
		tmp = -x * z;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9e-135:
		tmp = z * t
	elif t <= 3.8e-65:
		tmp = -x * z
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e-135)
		tmp = Float64(z * t);
	elseif (t <= 3.8e-65)
		tmp = Float64(Float64(-x) * z);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9e-135)
		tmp = z * t;
	elseif (t <= 3.8e-65)
		tmp = -x * z;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e-135], N[(z * t), $MachinePrecision], If[LessEqual[t, 3.8e-65], N[((-x) * z), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.99999999999999975e-135 or 3.8000000000000002e-65 < t

   1. Initial program 94.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 25.1%

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

   if -8.99999999999999975e-135 < t < 3.8000000000000002e-65

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 23.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 20.2%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-135}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.05e+49) (fma (- x) z x) (fma (- t x) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.05e+49) {
		tmp = fma(-x, z, x);
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.05e+49)
		tmp = fma(Float64(-x), z, x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.05e+49], N[((-x) * z + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.05e49

   1. Initial program 94.8%

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

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 50.4%

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

   if 2.05e49 < y

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 10: 26.8% accurate, 26.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 60.0

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

5. Applied rewrites 60.0%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 26.3%

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

Alternative 11: 17.3% accurate, 39.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 60.0

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

5. Applied rewrites 60.0%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 18.9%

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

9. Final simplification 18.9%

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

Developer Target 1: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024235 
(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))))))