SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.6% → 97.9%

Time: 13.1s

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: 93.6% 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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 77.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t + \frac{t \cdot t}{x}}{x}}{x}}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.2e+124)
   (fma (* y (- (/ t y) (tanh (/ x y)))) z x)
   (if (<= x 1.35e+59)
     (fma (* y (- (tanh (/ t y)) (/ x y))) z x)
     (fma z (/ 1.0 (/ (- -1.0 (/ (+ t (/ (* t t) x)) x)) x)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.2e+124) {
		tmp = fma((y * ((t / y) - tanh((x / y)))), z, x);
	} else if (x <= 1.35e+59) {
		tmp = fma((y * (tanh((t / y)) - (x / y))), z, x);
	} else {
		tmp = fma(z, (1.0 / ((-1.0 - ((t + ((t * t) / x)) / x)) / x)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.2e+124)
		tmp = fma(Float64(y * Float64(Float64(t / y) - tanh(Float64(x / y)))), z, x);
	elseif (x <= 1.35e+59)
		tmp = fma(Float64(y * Float64(tanh(Float64(t / y)) - Float64(x / y))), z, x);
	else
		tmp = fma(z, Float64(1.0 / Float64(Float64(-1.0 - Float64(Float64(t + Float64(Float64(t * t) / x)) / x)) / x)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.2e+124], N[(N[(y * N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[x, 1.35e+59], N[(N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], N[(z * N[(1.0 / N[(N[(-1.0 - N[(N[(t + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2000000000000001e124

   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 t around 0

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

      1. lower-/.f64 80.3

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

   7. Applied rewrites 80.3%

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

   if -2.2000000000000001e124 < x < 1.3500000000000001e59

   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 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 87.4

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

   7. Applied rewrites 87.4%

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

   if 1.3500000000000001e59 < x

   1. Initial program 98.1%

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

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

      3. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 41.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 62.6%

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

   11. Taylor expanded in x around -inf

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

   13. Applied rewrites 82.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t + \frac{t \cdot t}{x}}{x}}{x}}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t + \frac{t \cdot t}{x}}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 7e-109)
   (fma z (/ 1.0 (/ (- -1.0 (/ (+ t (/ (* t t) x)) x)) x)) x)
   (fma (* y (- (/ t y) (tanh (/ x y)))) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7e-109) {
		tmp = fma(z, (1.0 / ((-1.0 - ((t + ((t * t) / x)) / x)) / x)), x);
	} else {
		tmp = fma((y * ((t / y) - tanh((x / y)))), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7e-109)
		tmp = fma(z, Float64(1.0 / Float64(Float64(-1.0 - Float64(Float64(t + Float64(Float64(t * t) / x)) / x)) / x)), x);
	else
		tmp = fma(Float64(y * Float64(Float64(t / y) - tanh(Float64(x / y)))), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 7e-109], N[(z * N[(1.0 / N[(N[(-1.0 - N[(N[(t + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y * N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7e-109

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 41.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 55.9%

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

   11. Taylor expanded in x around -inf

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

   13. Applied rewrites 63.0%

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

   if 7e-109 < y

   1. Initial program 90.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 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 t around 0

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

      1. lower-/.f64 79.4

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

   7. Applied rewrites 79.4%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t + \frac{t \cdot t}{x}}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 90000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t + \frac{t \cdot t}{x}}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 90000000.0)
   (fma z (/ 1.0 (/ (- -1.0 (/ (+ t (/ (* t t) x)) x)) x)) x)
   (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 90000000.0) {
		tmp = fma(z, (1.0 / ((-1.0 - ((t + ((t * t) / x)) / x)) / x)), x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 90000000.0)
		tmp = fma(z, Float64(1.0 / Float64(Float64(-1.0 - Float64(Float64(t + Float64(Float64(t * t) / x)) / x)) / x)), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 90000000.0], N[(z * N[(1.0 / N[(N[(-1.0 - N[(N[(t + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9e7

   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 y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 40.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 54.1%

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

   11. Taylor expanded in x around -inf

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

   13. Applied rewrites 62.0%

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

   if 9e7 < y

   1. Initial program 86.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 90000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t + \frac{t \cdot t}{x}}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 90000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 90000000.0)
   (fma z (/ 1.0 (/ (- -1.0 (/ t x)) x)) x)
   (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 90000000.0) {
		tmp = fma(z, (1.0 / ((-1.0 - (t / x)) / x)), x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 90000000.0)
		tmp = fma(z, Float64(1.0 / Float64(Float64(-1.0 - Float64(t / x)) / x)), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 90000000.0], N[(z * N[(1.0 / N[(N[(-1.0 - N[(t / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9e7

   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 y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 40.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 59.9%

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

   if 9e7 < y

   1. Initial program 86.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 90000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{\frac{-1 - \frac{t}{x}}{x}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-103}:\\ \;\;\;\;\left(-t\right) \cdot \frac{x}{-t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.7e-103) (* (- t) (/ x (- t))) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.7e-103) {
		tmp = -t * (x / -t);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.7e-103)
		tmp = Float64(Float64(-t) * Float64(x / Float64(-t)));
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.7e-103], N[((-t) * N[(x / (-t)), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.70000000000000001e-103

   1. Initial program 97.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in t around -inf

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

   8. Applied rewrites 47.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 49.6%

       \[\leadsto \left(-t\right) \cdot \frac{x}{-t} \]

   if 1.70000000000000001e-103 < y

   1. Initial program 89.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 7: 62.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(t - x\right)\\ \mathbf{if}\;z \leq -1050000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(z, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- t x))))
   (if (<= z -1050000000.0) t_1 (if (<= z 1.25e+39) (fma z (- x) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (t - x);
	double tmp;
	if (z <= -1050000000.0) {
		tmp = t_1;
	} else if (z <= 1.25e+39) {
		tmp = fma(z, -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(t - x))
	tmp = 0.0
	if (z <= -1050000000.0)
		tmp = t_1;
	elseif (z <= 1.25e+39)
		tmp = fma(z, Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1050000000.0], t$95$1, If[LessEqual[z, 1.25e+39], N[(z * (-x) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e9 or 1.25000000000000004e39 < z

   1. Initial program 91.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 46.3%

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

   if -1.05e9 < z < 1.25000000000000004e39

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 73.5

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 85.0%

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

Alternative 8: 20.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{-105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x))))
   (if (<= x -1.32e-105) t_1 (if (<= x 1.35e-76) (* t z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * -x;
	double tmp;
	if (x <= -1.32e-105) {
		tmp = t_1;
	} else if (x <= 1.35e-76) {
		tmp = t * z;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = z * -x
    if (x <= (-1.32d-105)) then
        tmp = t_1
    else if (x <= 1.35d-76) then
        tmp = t * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * -x;
	double tmp;
	if (x <= -1.32e-105) {
		tmp = t_1;
	} else if (x <= 1.35e-76) {
		tmp = t * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * -x
	tmp = 0
	if x <= -1.32e-105:
		tmp = t_1
	elif x <= 1.35e-76:
		tmp = t * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(-x))
	tmp = 0.0
	if (x <= -1.32e-105)
		tmp = t_1;
	elseif (x <= 1.35e-76)
		tmp = Float64(t * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * -x;
	tmp = 0.0;
	if (x <= -1.32e-105)
		tmp = t_1;
	elseif (x <= 1.35e-76)
		tmp = t * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[x, -1.32e-105], t$95$1, If[LessEqual[x, 1.35e-76], N[(t * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32000000000000006e-105 or 1.35e-76 < x

   1. Initial program 96.4%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 29.7%

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

   8. Taylor expanded in z around -inf

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

   10. Applied rewrites 22.1%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 19.3%

       \[\leadsto z \cdot \left(-x\right) \]

   if -1.32000000000000006e-105 < x < 1.35e-76

   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. lower-fma.f64 N/A

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

      3. lower--.f64 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 31.7%

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

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-105}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.1% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 66000000:\\ \;\;\;\;\mathsf{fma}\left(z, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 66000000.0) (fma z (- x) x) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 66000000.0) {
		tmp = fma(z, -x, x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 66000000.0)
		tmp = fma(z, Float64(-x), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 66000000.0], N[(z * (-x) + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.6e7

   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 y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.3%

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

   if 6.6e7 < y

   1. Initial program 86.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 79.6

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

   5. Applied rewrites 79.6%

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

Alternative 10: 26.4% accurate, 26.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower--.f64 60.7

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

5. Applied rewrites 60.7%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 26.6%

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

Alternative 11: 16.6% accurate, 39.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower--.f64 60.7

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

5. Applied rewrites 60.7%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 16.9%

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

9. Final simplification 16.9%

   \[\leadsto t \cdot z \]
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 2024233 
(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))))))