SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.3% → 97.0%

Time: 9.8s

Alternatives: 9

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

   1. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tanh.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-tanh.f64 N/A

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

   9. +-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

3. Applied rewrites 97.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tanh.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-tanh.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

5. Applied rewrites 97.0%

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

Alternative 2: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, z \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x - \left(z \cdot y\right) \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (* z (tanh (/ t y))) x)))
   (if (<= t -2.4e-28)
     t_1
     (if (<= t 2.75e+23) (fma z t (- x (* (* z y) (tanh (/ x y))))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z * tanh((t / y))), x);
	double tmp;
	if (t <= -2.4e-28) {
		tmp = t_1;
	} else if (t <= 2.75e+23) {
		tmp = fma(z, t, (x - ((z * y) * tanh((x / y)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z * tanh(Float64(t / y))), x)
	tmp = 0.0
	if (t <= -2.4e-28)
		tmp = t_1;
	elseif (t <= 2.75e+23)
		tmp = fma(z, t, Float64(x - Float64(Float64(z * y) * tanh(Float64(x / y)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -2.4e-28], t$95$1, If[LessEqual[t, 2.75e+23], N[(z * t + N[(x - N[(N[(z * y), $MachinePrecision] * N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4000000000000002e-28 or 2.75000000000000002e23 < t

   1. Initial program 96.3%

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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tanh.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-tanh.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-sub N/A

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

      3. rec-exp N/A

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

      4. rec-exp N/A

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

      5. tanh-def-a N/A

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

      6. lift-tanh.f64 N/A

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

      7. lift-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

   if -2.4000000000000002e-28 < t < 2.75000000000000002e23

   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. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tanh.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-tanh.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 95.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tanh.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-tanh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 82.6%

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

Alternative 3: 85.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot t\_1, x\right)\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-114}:\\ \;\;\;\;x - \left(z \cdot y\right) \cdot \tanh \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))))
   (if (<= t -6.4e-29)
     (fma y (* z t_1) x)
     (if (<= t 3.1e-114)
       (- x (* (* z y) (tanh (/ x y))))
       (+ x (* (* y z) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double tmp;
	if (t <= -6.4e-29) {
		tmp = fma(y, (z * t_1), x);
	} else if (t <= 3.1e-114) {
		tmp = x - ((z * y) * tanh((x / y)));
	} else {
		tmp = x + ((y * z) * t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	tmp = 0.0
	if (t <= -6.4e-29)
		tmp = fma(y, Float64(z * t_1), x);
	elseif (t <= 3.1e-114)
		tmp = Float64(x - Float64(Float64(z * y) * tanh(Float64(x / y))));
	else
		tmp = Float64(x + Float64(Float64(y * z) * t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -6.4e-29], N[(y * N[(z * t$95$1), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 3.1e-114], N[(x - N[(N[(z * y), $MachinePrecision] * N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.400000000000001e-29

   1. Initial program 95.6%

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

      1. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tanh.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-tanh.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-sub N/A

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

      3. rec-exp N/A

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

      4. rec-exp N/A

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

      5. tanh-def-a N/A

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

      6. lift-tanh.f64 N/A

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

      7. lift-/.f64 88.0

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

   6. Applied rewrites 88.0%

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

   if -6.400000000000001e-29 < t < 3.1e-114

   1. Initial program 89.8%

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

   2. Taylor expanded in t around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. lower--.f64 N/A

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

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

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

      4. associate-*r* N/A

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

   4. Applied rewrites 80.4%

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

   if 3.1e-114 < t

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-sub N/A

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

      3. rec-exp N/A

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

      4. rec-exp N/A

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

      5. tanh-def-a N/A

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

      6. lift-tanh.f64 N/A

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

      7. lift-/.f64 84.5

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

   4. Applied rewrites 84.5%

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

Alternative 4: 83.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, x, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.6e+86)
   (fma (- t x) z x)
   (if (<= y 1.1e+57)
     (fma y (* z (tanh (/ t y))) x)
     (fma (- 1.0 z) x (* t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.6e+86) {
		tmp = fma((t - x), z, x);
	} else if (y <= 1.1e+57) {
		tmp = fma(y, (z * tanh((t / y))), x);
	} else {
		tmp = fma((1.0 - z), x, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.6e+86)
		tmp = fma(Float64(t - x), z, x);
	elseif (y <= 1.1e+57)
		tmp = fma(y, Float64(z * tanh(Float64(t / y))), x);
	else
		tmp = fma(Float64(1.0 - z), x, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.6e+86], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[y, 1.1e+57], N[(y * N[(z * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * x + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.60000000000000005e86

   1. Initial program 83.7%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

         \[\leadsto \left(1 \cdot \left(t - x\right)\right) \cdot z + x \]

      4. metadata-eval N/A

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

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

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

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

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

      7. lower-fma.f64 N/A

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

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

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower--.f64 N/A

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

      14. *-rgt-identity 86.2

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

   4. Applied rewrites 86.2%

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

   if -3.60000000000000005e86 < y < 1.1e57

   1. Initial program 99.4%

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

      1. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tanh.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-tanh.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-sub N/A

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

      3. rec-exp N/A

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

      4. rec-exp N/A

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

      5. tanh-def-a N/A

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

      6. lift-tanh.f64 N/A

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

      7. lift-/.f64 84.8

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

   6. Applied rewrites 84.8%

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

   if 1.1e57 < y

   1. Initial program 83.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 72.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 84.7

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

   7. Applied rewrites 84.7%

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

Alternative 5: 78.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, x, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8e+18)
   (fma (- t x) z x)
   (if (<= y 2.4e+28) x (fma (- 1.0 z) x (* t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8e+18) {
		tmp = fma((t - x), z, x);
	} else if (y <= 2.4e+28) {
		tmp = x;
	} else {
		tmp = fma((1.0 - z), x, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8e+18)
		tmp = fma(Float64(t - x), z, x);
	elseif (y <= 2.4e+28)
		tmp = x;
	else
		tmp = fma(Float64(1.0 - z), x, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8e+18], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[y, 2.4e+28], x, N[(N[(1.0 - z), $MachinePrecision] * x + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8e18

   1. Initial program 86.5%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

         \[\leadsto \left(1 \cdot \left(t - x\right)\right) \cdot z + x \]

      4. metadata-eval N/A

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

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

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

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

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

      7. lower-fma.f64 N/A

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

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

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower--.f64 N/A

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

      14. *-rgt-identity 80.2

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

   4. Applied rewrites 80.2%

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

   if -8e18 < y < 2.39999999999999981e28

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 75.4%

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

   if 2.39999999999999981e28 < y

   1. Initial program 84.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 82.0

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

   7. Applied rewrites 82.0%

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

Alternative 6: 78.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{if}\;y \leq -8 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- t x) z x)))
   (if (<= y -8e+18) t_1 (if (<= y 2.4e+28) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((t - x), z, x);
	double tmp;
	if (y <= -8e+18) {
		tmp = t_1;
	} else if (y <= 2.4e+28) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(t - x), z, x)
	tmp = 0.0
	if (y <= -8e+18)
		tmp = t_1;
	elseif (y <= 2.4e+28)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[y, -8e+18], t$95$1, If[LessEqual[y, 2.4e+28], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8e18 or 2.39999999999999981e28 < y

   1. Initial program 85.6%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

         \[\leadsto \left(1 \cdot \left(t - x\right)\right) \cdot z + x \]

      4. metadata-eval N/A

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

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

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

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

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

      7. lower-fma.f64 N/A

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

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

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower--.f64 N/A

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

      14. *-rgt-identity 81.3

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

   4. Applied rewrites 81.3%

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

   if -8e18 < y < 2.39999999999999981e28

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 75.4%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.8e+29) (fma t z x) (if (<= y 3.3e+28) x (fma t z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.8e+29) {
		tmp = fma(t, z, x);
	} else if (y <= 3.3e+28) {
		tmp = x;
	} else {
		tmp = fma(t, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.8e+29)
		tmp = fma(t, z, x);
	elseif (y <= 3.3e+28)
		tmp = x;
	else
		tmp = fma(t, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.8e+29], N[(t * z + x), $MachinePrecision], If[LessEqual[y, 3.3e+28], x, N[(t * z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.79999999999999937e29 or 3.3e28 < y

   1. Initial program 85.4%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

         \[\leadsto \left(1 \cdot \left(t - x\right)\right) \cdot z + x \]

      4. metadata-eval N/A

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

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

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

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

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

      7. lower-fma.f64 N/A

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

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

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower--.f64 N/A

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

      14. *-rgt-identity 82.0

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.8%

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

   if -7.79999999999999937e29 < y < 3.3e28

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 75.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y)))))))
   (if (<= t_1 (- INFINITY)) (* z t) (if (<= t_1 INFINITY) x (* z t)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 47.6%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. div-sub N/A

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

      4. rec-exp N/A

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

      5. rec-exp N/A

         \[\leadsto \left(y \cdot z\right) \cdot \frac{e^{\frac{t}{y}} - e^{\mathsf{neg}\left(\frac{t}{y}\right)}}{e^{\frac{t}{y}} + e^{\mathsf{neg}\left(\frac{t}{y}\right)}} \]

      6. tanh-def-a N/A

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto z \cdot t \]

      2. lower-*.f64 52.9

         \[\leadsto z \cdot t \]

   7. Applied rewrites 52.9%

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

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

   1. Initial program 96.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 65.1%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 60.8% accurate, 34.7× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 93.3%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 60.8%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64
  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))