SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.6% → 97.1%

Time: 13.1s

Alternatives: 8

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: 93.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq 1.45 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, t_1 - \tanh \left(\frac{x}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot t_1 - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))))
   (if (<= y 1.45e+138)
     (fma (* y z) (- t_1 (tanh (/ x y))) x)
     (+ x (* z (- (* y t_1) x))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double tmp;
	if (y <= 1.45e+138) {
		tmp = fma((y * z), (t_1 - tanh((x / y))), x);
	} else {
		tmp = x + (z * ((y * t_1) - x));
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	tmp = 0.0
	if (y <= 1.45e+138)
		tmp = fma(Float64(y * z), Float64(t_1 - tanh(Float64(x / y))), x);
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y * t_1) - x)));
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.45e+138], N[(N[(y * z), $MachinePrecision] * N[(t$95$1 - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(z * N[(N[(y * t$95$1), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45000000000000005e138

   1. Initial program 95.5%

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

      1. +-commutative 95.5%

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

      2. fma-def 95.5%

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

   3. Simplified 95.5%

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

   if 1.45000000000000005e138 < y

   1. Initial program 80.9%

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

   2. Taylor expanded in x around 0 55.5%

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

      1. +-commutative 55.5%

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

   4. Simplified 96.2%

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

      1. add-sqr-sqrt 56.2%

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

      2. sqrt-unprod 79.2%

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

      3. pow2 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. unpow2 79.2%

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

      2. rem-sqrt-square 78.9%

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

   8. Simplified 78.9%

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

   9. Taylor expanded in z around 0 78.9%

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

       1. expm1-log1p-u 77.3%

          \[\leadsto x + z \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left|\tanh \left(\frac{t}{y}\right)\right|\right)\right)} - x\right) \]

       2. expm1-udef 77.4%

          \[\leadsto x + z \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left|\tanh \left(\frac{t}{y}\right)\right|\right)} - 1\right)} - x\right) \]

       3. add-sqr-sqrt 54.6%

          \[\leadsto x + z \cdot \left(\left(e^{\mathsf{log1p}\left(y \cdot \left|\color{blue}{\sqrt{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt{\tanh \left(\frac{t}{y}\right)}}\right|\right)} - 1\right) - x\right) \]

       4. fabs-sqr 54.6%

          \[\leadsto x + z \cdot \left(\left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt{\tanh \left(\frac{t}{y}\right)}\right)}\right)} - 1\right) - x\right) \]

       5. add-sqr-sqrt 61.3%

          \[\leadsto x + z \cdot \left(\left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\tanh \left(\frac{t}{y}\right)}\right)} - 1\right) - x\right) \]

   11. Applied egg-rr 61.3%

       \[\leadsto x + z \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \tanh \left(\frac{t}{y}\right)\right)} - 1\right)} - x\right) \]
   12. Step-by-step derivation

       1. expm1-def 66.6%

          \[\leadsto x + z \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\right)} - x\right) \]

       2. expm1-log1p 96.2%

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

   13. Simplified 96.2%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right) - x\right)\\ \end{array} \]

Alternative 2: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq 4 \cdot 10^{+138}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(t_1 - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot t_1 - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))))
   (if (<= y 4e+138)
     (+ x (* (* y z) (- t_1 (tanh (/ x y)))))
     (+ x (* z (- (* y t_1) x))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double tmp;
	if (y <= 4e+138) {
		tmp = x + ((y * z) * (t_1 - tanh((x / y))));
	} else {
		tmp = x + (z * ((y * t_1) - x));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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) :: t_1
    real(8) :: tmp
    t_1 = tanh((t / y))
    if (y <= 4d+138) then
        tmp = x + ((y * z) * (t_1 - tanh((x / y))))
    else
        tmp = x + (z * ((y * t_1) - x))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.tanh((t / y));
	double tmp;
	if (y <= 4e+138) {
		tmp = x + ((y * z) * (t_1 - Math.tanh((x / y))));
	} else {
		tmp = x + (z * ((y * t_1) - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	t_1 = math.tanh((t / y))
	tmp = 0
	if y <= 4e+138:
		tmp = x + ((y * z) * (t_1 - math.tanh((x / y))))
	else:
		tmp = x + (z * ((y * t_1) - x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	tmp = 0.0
	if (y <= 4e+138)
		tmp = Float64(x + Float64(Float64(y * z) * Float64(t_1 - tanh(Float64(x / y)))));
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y * t_1) - x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	t_1 = tanh((t / y));
	tmp = 0.0;
	if (y <= 4e+138)
		tmp = x + ((y * z) * (t_1 - tanh((x / y))));
	else
		tmp = x + (z * ((y * t_1) - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4e+138], N[(x + N[(N[(y * z), $MachinePrecision] * N[(t$95$1 - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(N[(y * t$95$1), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000001e138

   1. Initial program 95.5%

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

   if 4.0000000000000001e138 < y

   1. Initial program 80.9%

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

   2. Taylor expanded in x around 0 55.5%

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

      1. +-commutative 55.5%

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

   4. Simplified 96.2%

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

      1. add-sqr-sqrt 56.2%

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

      2. sqrt-unprod 79.2%

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

      3. pow2 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. unpow2 79.2%

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

      2. rem-sqrt-square 78.9%

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

   8. Simplified 78.9%

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

   9. Taylor expanded in z around 0 78.9%

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

       1. expm1-log1p-u 77.3%

          \[\leadsto x + z \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left|\tanh \left(\frac{t}{y}\right)\right|\right)\right)} - x\right) \]

       2. expm1-udef 77.4%

          \[\leadsto x + z \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left|\tanh \left(\frac{t}{y}\right)\right|\right)} - 1\right)} - x\right) \]

       3. add-sqr-sqrt 54.6%

          \[\leadsto x + z \cdot \left(\left(e^{\mathsf{log1p}\left(y \cdot \left|\color{blue}{\sqrt{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt{\tanh \left(\frac{t}{y}\right)}}\right|\right)} - 1\right) - x\right) \]

       4. fabs-sqr 54.6%

          \[\leadsto x + z \cdot \left(\left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt{\tanh \left(\frac{t}{y}\right)}\right)}\right)} - 1\right) - x\right) \]

       5. add-sqr-sqrt 61.3%

          \[\leadsto x + z \cdot \left(\left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\tanh \left(\frac{t}{y}\right)}\right)} - 1\right) - x\right) \]

   11. Applied egg-rr 61.3%

       \[\leadsto x + z \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \tanh \left(\frac{t}{y}\right)\right)} - 1\right)} - x\right) \]
   12. Step-by-step derivation

       1. expm1-def 66.6%

          \[\leadsto x + z \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\right)} - x\right) \]

       2. expm1-log1p 96.2%

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

   13. Simplified 96.2%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+138}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right) - x\right)\\ \end{array} \]

Alternative 3: 87.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq 6.4 \cdot 10^{+16}:\\ \;\;\;\;x + y \cdot \left(z \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot t_1 - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))))
   (if (<= y 6.4e+16) (+ x (* y (* z t_1))) (+ x (* z (- (* y t_1) x))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x + (y * (z * t_1));
	} else {
		tmp = x + (z * ((y * t_1) - x));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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) :: t_1
    real(8) :: tmp
    t_1 = tanh((t / y))
    if (y <= 6.4d+16) then
        tmp = x + (y * (z * t_1))
    else
        tmp = x + (z * ((y * t_1) - x))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.tanh((t / y));
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x + (y * (z * t_1));
	} else {
		tmp = x + (z * ((y * t_1) - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	t_1 = math.tanh((t / y))
	tmp = 0
	if y <= 6.4e+16:
		tmp = x + (y * (z * t_1))
	else:
		tmp = x + (z * ((y * t_1) - x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	tmp = 0.0
	if (y <= 6.4e+16)
		tmp = Float64(x + Float64(y * Float64(z * t_1)));
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y * t_1) - x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	t_1 = tanh((t / y));
	tmp = 0.0;
	if (y <= 6.4e+16)
		tmp = x + (y * (z * t_1));
	else
		tmp = x + (z * ((y * t_1) - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 6.4e+16], N[(x + N[(y * N[(z * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(N[(y * t$95$1), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4e16

   1. Initial program 95.3%

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

   2. Taylor expanded in x around 0 26.0%

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

      1. associate-/r* 26.0%

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

      2. div-sub 26.0%

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

      3. rec-exp 26.0%

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

      4. rec-exp 26.0%

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

      5. tanh-def-a 81.1%

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

   4. Simplified 81.1%

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

   if 6.4e16 < y

   1. Initial program 89.4%

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

   2. Taylor expanded in x around 0 47.5%

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

      1. +-commutative 47.5%

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

   4. Simplified 84.6%

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

      1. add-sqr-sqrt 53.4%

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

      2. sqrt-unprod 73.3%

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

      3. pow2 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. unpow2 73.3%

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

      2. rem-sqrt-square 73.2%

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

   8. Simplified 73.2%

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

   9. Taylor expanded in z around 0 73.2%

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

       1. expm1-log1p-u 72.1%

          \[\leadsto x + z \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left|\tanh \left(\frac{t}{y}\right)\right|\right)\right)} - x\right) \]

       2. expm1-udef 72.1%

          \[\leadsto x + z \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left|\tanh \left(\frac{t}{y}\right)\right|\right)} - 1\right)} - x\right) \]

       3. add-sqr-sqrt 52.4%

          \[\leadsto x + z \cdot \left(\left(e^{\mathsf{log1p}\left(y \cdot \left|\color{blue}{\sqrt{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt{\tanh \left(\frac{t}{y}\right)}}\right|\right)} - 1\right) - x\right) \]

       4. fabs-sqr 52.4%

          \[\leadsto x + z \cdot \left(\left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt{\tanh \left(\frac{t}{y}\right)}\right)}\right)} - 1\right) - x\right) \]

       5. add-sqr-sqrt 59.2%

          \[\leadsto x + z \cdot \left(\left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\tanh \left(\frac{t}{y}\right)}\right)} - 1\right) - x\right) \]

   11. Applied egg-rr 59.2%

       \[\leadsto x + z \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \tanh \left(\frac{t}{y}\right)\right)} - 1\right)} - x\right) \]
   12. Step-by-step derivation

       1. expm1-def 61.6%

          \[\leadsto x + z \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\right)} - x\right) \]

       2. expm1-log1p 84.6%

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

   13. Simplified 84.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+16}:\\ \;\;\;\;x + y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right) - x\right)\\ \end{array} \]

Alternative 4: 85.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+145}:\\ \;\;\;\;x + y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 5.2e+145) (+ x (* y (* z (tanh (/ t y))))) (+ x (* z (- t x)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.2e+145) {
		tmp = x + (y * (z * tanh((t / y))));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 (y <= 5.2d+145) then
        tmp = x + (y * (z * tanh((t / y))))
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.2e+145) {
		tmp = x + (y * (z * Math.tanh((t / y))));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 5.2e+145:
		tmp = x + (y * (z * math.tanh((t / y))))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.2e+145)
		tmp = Float64(x + Float64(y * Float64(z * tanh(Float64(t / y)))));
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 5.2e+145)
		tmp = x + (y * (z * tanh((t / y))));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 5.2e+145], N[(x + N[(y * N[(z * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000005e145

   1. Initial program 95.5%

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

   2. Taylor expanded in x around 0 26.5%

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

      1. associate-/r* 26.5%

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

      2. div-sub 26.5%

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

      3. rec-exp 26.5%

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

      4. rec-exp 26.5%

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

      5. tanh-def-a 79.3%

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

   4. Simplified 79.3%

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

   if 5.20000000000000005e145 < y

   1. Initial program 79.2%

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

   2. Taylor expanded in y around inf 87.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+145}:\\ \;\;\;\;x + y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 5: 69.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+189} \lor \neg \left(y \leq 4.1 \cdot 10^{+301}\right):\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.65e+17)
   x
   (if (or (<= y 3.55e+189) (not (<= y 4.1e+301)))
     (- x (* z x))
     (+ x (* z t)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.65e+17) {
		tmp = x;
	} else if ((y <= 3.55e+189) || !(y <= 4.1e+301)) {
		tmp = x - (z * x);
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 (y <= 2.65d+17) then
        tmp = x
    else if ((y <= 3.55d+189) .or. (.not. (y <= 4.1d+301))) then
        tmp = x - (z * x)
    else
        tmp = x + (z * t)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.65e+17) {
		tmp = x;
	} else if ((y <= 3.55e+189) || !(y <= 4.1e+301)) {
		tmp = x - (z * x);
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 2.65e+17:
		tmp = x
	elif (y <= 3.55e+189) or not (y <= 4.1e+301):
		tmp = x - (z * x)
	else:
		tmp = x + (z * t)
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.65e+17)
		tmp = x;
	elseif ((y <= 3.55e+189) || !(y <= 4.1e+301))
		tmp = Float64(x - Float64(z * x));
	else
		tmp = Float64(x + Float64(z * t));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.65e+17)
		tmp = x;
	elseif ((y <= 3.55e+189) || ~((y <= 4.1e+301)))
		tmp = x - (z * x);
	else
		tmp = x + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 2.65e+17], x, If[Or[LessEqual[y, 3.55e+189], N[Not[LessEqual[y, 4.1e+301]], $MachinePrecision]], N[(x - N[(z * x), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.65e17

   1. Initial program 95.3%

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

      1. +-commutative 95.3%

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

      2. fma-def 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in y around 0 66.7%

      \[\leadsto \color{blue}{x} \]

   if 2.65e17 < y < 3.55e189 or 4.0999999999999998e301 < y

   1. Initial program 95.0%

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

   2. Taylor expanded in x around 0 48.1%

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

      1. +-commutative 48.1%

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

   4. Simplified 81.2%

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

   5. Taylor expanded in x around inf 64.3%

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

      1. distribute-lft-in 64.3%

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

      2. *-rgt-identity 64.3%

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

      3. mul-1-neg 64.3%

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

      4. distribute-rgt-neg-in 64.3%

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

      5. unsub-neg 64.3%

         \[\leadsto \color{blue}{x - x \cdot z} \]

      6. *-commutative 64.3%

         \[\leadsto x - \color{blue}{z \cdot x} \]

   7. Simplified 64.3%

      \[\leadsto \color{blue}{x - z \cdot x} \]

   if 3.55e189 < y < 4.0999999999999998e301

   1. Initial program 74.5%

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

   2. Taylor expanded in x around 0 28.3%

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

      1. associate-/r* 28.3%

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

      2. div-sub 28.3%

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

      3. rec-exp 28.3%

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

      4. rec-exp 28.3%

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

      5. tanh-def-a 61.5%

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

   4. Simplified 61.5%

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

   5. Taylor expanded in y around inf 74.6%

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

      1. +-commutative 74.6%

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

      2. *-commutative 74.6%

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

   7. Simplified 74.6%

      \[\leadsto \color{blue}{z \cdot t + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+189} \lor \neg \left(y \leq 4.1 \cdot 10^{+301}\right):\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \]

Alternative 6: 78.5% accurate, 23.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.6e+15) x (+ x (* z (- t x)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.6e+15) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 (y <= 6.6d+15) then
        tmp = x
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.6e+15) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 6.6e+15:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.6e+15)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 6.6e+15)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 6.6e+15], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.6e15

   1. Initial program 95.3%

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

      1. +-commutative 95.3%

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

      2. fma-def 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in y around 0 66.7%

      \[\leadsto \color{blue}{x} \]

   if 6.6e15 < y

   1. Initial program 89.4%

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

   2. Taylor expanded in y around inf 71.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 7: 70.8% accurate, 30.2× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+90}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t) :precision binary64 (if (<= y 4.2e+90) x (+ x (* z t))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.2e+90) {
		tmp = x;
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 (y <= 4.2d+90) then
        tmp = x
    else
        tmp = x + (z * t)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.2e+90) {
		tmp = x;
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 4.2e+90:
		tmp = x
	else:
		tmp = x + (z * t)
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.2e+90)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * t));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.2e+90)
		tmp = x;
	else
		tmp = x + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 4.2e+90], x, N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.19999999999999961e90

   1. Initial program 95.3%

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

      1. +-commutative 95.3%

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

      2. fma-def 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in y around 0 64.9%

      \[\leadsto \color{blue}{x} \]

   if 4.19999999999999961e90 < y

   1. Initial program 85.1%

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

   2. Taylor expanded in x around 0 38.6%

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

      1. associate-/r* 38.6%

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

      2. div-sub 38.6%

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

      3. rec-exp 38.6%

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

      4. rec-exp 38.6%

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

      5. tanh-def-a 69.7%

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

   4. Simplified 69.7%

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

   5. Taylor expanded in y around inf 66.1%

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

      1. +-commutative 66.1%

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

      2. *-commutative 66.1%

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

   7. Simplified 66.1%

      \[\leadsto \color{blue}{z \cdot t + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+90}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \]

Alternative 8: 61.0% accurate, 213.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ x \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t) :precision binary64 x)

C

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

Fortran

NOTE: y should be positive before calling this function
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
end function

Java

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

Python

y = abs(y)
def code(x, y, z, t):
	return x

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := x

Derivation

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. Step-by-step derivation

   1. +-commutative 94.0%

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

   2. fma-def 94.1%

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

3. Simplified 94.1%

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

4. Taylor expanded in y around 0 62.1%

   \[\leadsto \color{blue}{x} \]

5. Final simplification 62.1%

   \[\leadsto x \]

Developer target: 97.2% 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 2023287 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :herbie-target
  (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))

  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))