SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.3% → 97.3%

Time: 3.4s

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: 97.3% 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. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

3. Applied rewrites 93.3%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tanh.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-tanh.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tanh.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tanh.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift--.f64 97.3

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

5. Applied rewrites 97.3%

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

Alternative 2: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ t_2 := x + \left(y \cdot z\right) \cdot \left(t\_1 - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{elif}\;t\_2 \leq 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, t\_1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) + 1, x, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))) (t_2 (+ x (* (* y z) (- t_1 (tanh (/ x y)))))))
   (if (<= t_2 (- INFINITY))
     (* (- t x) z)
     (if (<= t_2 1e+300) (fma (* z y) t_1 x) (fma (+ (- z) 1.0) x (* z t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double t_2 = x + ((y * z) * (t_1 - tanh((x / y))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t - x) * z;
	} else if (t_2 <= 1e+300) {
		tmp = fma((z * y), t_1, x);
	} else {
		tmp = fma((-z + 1.0), x, (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	t_2 = Float64(x + Float64(Float64(y * z) * Float64(t_1 - tanh(Float64(x / y)))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t - x) * z);
	elseif (t_2 <= 1e+300)
		tmp = fma(Float64(z * y), t_1, x);
	else
		tmp = fma(Float64(Float64(-z) + 1.0), x, Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * z), $MachinePrecision] * N[(t$95$1 - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$2, 1e+300], N[(N[(z * y), $MachinePrecision] * t$95$1 + x), $MachinePrecision], N[(N[((-z) + 1.0), $MachinePrecision] * x + N[(z * t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 13.3

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

   10. Applied rewrites 13.3%

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

   11. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 26.5

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

   13. Applied rewrites 26.5%

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.0000000000000001e300

   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 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 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 80.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 80.0

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

   6. Applied rewrites 80.0%

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

   if 1.0000000000000001e300 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

Alternative 3: 78.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{elif}\;y \leq 430000000000:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) + 1, x, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.7e+58)
   (fma (- t x) z x)
   (if (<= y 430000000000.0) (* (- x) -1.0) (fma (+ (- z) 1.0) x (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.7e+58) {
		tmp = fma((t - x), z, x);
	} else if (y <= 430000000000.0) {
		tmp = -x * -1.0;
	} else {
		tmp = fma((-z + 1.0), x, (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.7e+58)
		tmp = fma(Float64(t - x), z, x);
	elseif (y <= 430000000000.0)
		tmp = Float64(Float64(-x) * -1.0);
	else
		tmp = fma(Float64(Float64(-z) + 1.0), x, Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.7e+58], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[y, 430000000000.0], N[((-x) * -1.0), $MachinePrecision], N[(N[((-z) + 1.0), $MachinePrecision] * x + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7000000000000001e58

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   if -2.7000000000000001e58 < y < 4.3e11

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 61.0%

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

   if 4.3e11 < y

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

Alternative 4: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 430000000000:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- t x) z x)))
   (if (<= y -2.7e+58) t_1 (if (<= y 430000000000.0) (* (- x) -1.0) 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 <= -2.7e+58) {
		tmp = t_1;
	} else if (y <= 430000000000.0) {
		tmp = -x * -1.0;
	} 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 <= -2.7e+58)
		tmp = t_1;
	elseif (y <= 430000000000.0)
		tmp = Float64(Float64(-x) * -1.0);
	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, -2.7e+58], t$95$1, If[LessEqual[y, 430000000000.0], N[((-x) * -1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7000000000000001e58 or 4.3e11 < y

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   if -2.7000000000000001e58 < y < 4.3e11

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 61.0%

       \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ t_2 := x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+300}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) z))
        (t_2 (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y)))))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 1e+300) (* (- x) -1.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * z;
	double t_2 = x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+300) {
		tmp = -x * -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * z;
	double t_2 = x + ((y * z) * (Math.tanh((t / y)) - Math.tanh((x / y))));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 1e+300) {
		tmp = -x * -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t - x) * z
	t_2 = x + ((y * z) * (math.tanh((t / y)) - math.tanh((x / y))))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 1e+300:
		tmp = -x * -1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * z)
	t_2 = Float64(x + Float64(Float64(y * z) * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+300)
		tmp = Float64(Float64(-x) * -1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t - x) * z;
	t_2 = x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 1e+300)
		tmp = -x * -1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = 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$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+300], N[((-x) * -1.0), $MachinePrecision], t$95$1]]]]

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 1.0000000000000001e300 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 13.3

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

   10. Applied rewrites 13.3%

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

   11. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 26.5

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

   13. Applied rewrites 26.5%

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.0000000000000001e300

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 61.0%

       \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.9 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x\right)\\ \mathbf{elif}\;y \leq 450000000000:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.9e+59)
   (fma t z x)
   (if (<= y 450000000000.0) (* (- x) -1.0) (fma t z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.9e+59) {
		tmp = fma(t, z, x);
	} else if (y <= 450000000000.0) {
		tmp = -x * -1.0;
	} else {
		tmp = fma(t, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.9e+59)
		tmp = fma(t, z, x);
	elseif (y <= 450000000000.0)
		tmp = Float64(Float64(-x) * -1.0);
	else
		tmp = fma(t, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.9e+59], N[(t * z + x), $MachinePrecision], If[LessEqual[y, 450000000000.0], N[((-x) * -1.0), $MachinePrecision], N[(t * z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.90000000000000038e59 or 4.5e11 < y

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 59.0%

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

   if -5.90000000000000038e59 < y < 4.5e11

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 61.0%

       \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t\_1 \leq 10^{+300}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \end{array} \end{array} \]

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 1e+300) (* (- x) -1.0) (* (- x) z)))))

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 <= 1e+300) {
		tmp = -x * -1.0;
	} else {
		tmp = -x * z;
	}
	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 <= 1e+300) {
		tmp = -x * -1.0;
	} else {
		tmp = -x * z;
	}
	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 <= 1e+300:
		tmp = -x * -1.0
	else:
		tmp = -x * z
	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 <= 1e+300)
		tmp = Float64(Float64(-x) * -1.0);
	else
		tmp = Float64(Float64(-x) * z);
	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 <= 1e+300)
		tmp = -x * -1.0;
	else
		tmp = -x * z;
	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, 1e+300], N[((-x) * -1.0), $MachinePrecision], N[((-x) * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto z \cdot t \]

      2. lower-*.f64 17.1

         \[\leadsto z \cdot t \]

   7. Applied rewrites 17.1%

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.0000000000000001e300

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 61.0%

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

   if 1.0000000000000001e300 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 13.3

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

   10. Applied rewrites 13.3%

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

Alternative 8: 20.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot z\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+117}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x) z)))
   (if (<= x -1.85e+73) t_1 (if (<= x 5.4e+117) (* z t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -x * z;
	double tmp;
	if (x <= -1.85e+73) {
		tmp = t_1;
	} else if (x <= 5.4e+117) {
		tmp = z * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -x * z
    if (x <= (-1.85d+73)) then
        tmp = t_1
    else if (x <= 5.4d+117) then
        tmp = z * t
    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 = -x * z;
	double tmp;
	if (x <= -1.85e+73) {
		tmp = t_1;
	} else if (x <= 5.4e+117) {
		tmp = z * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -x * z
	tmp = 0
	if x <= -1.85e+73:
		tmp = t_1
	elif x <= 5.4e+117:
		tmp = z * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-x) * z)
	tmp = 0.0
	if (x <= -1.85e+73)
		tmp = t_1;
	elseif (x <= 5.4e+117)
		tmp = Float64(z * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -x * z;
	tmp = 0.0;
	if (x <= -1.85e+73)
		tmp = t_1;
	elseif (x <= 5.4e+117)
		tmp = z * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-x) * z), $MachinePrecision]}, If[LessEqual[x, -1.85e+73], t$95$1, If[LessEqual[x, 5.4e+117], N[(z * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.84999999999999987e73 or 5.4000000000000005e117 < x

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 13.3

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

   10. Applied rewrites 13.3%

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

   if -1.84999999999999987e73 < x < 5.4000000000000005e117

   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 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. lower-fma.f64 N/A

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

      4. lower--.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto z \cdot t \]

      2. lower-*.f64 17.1

         \[\leadsto z \cdot t \]

   7. Applied rewrites 17.1%

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

Alternative 9: 17.1% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Fortran

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 = z * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 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. lower-fma.f64 N/A

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

   4. lower--.f64 61.2

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

4. Applied rewrites 61.2%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto z \cdot t \]

   2. lower-*.f64 17.1

      \[\leadsto z \cdot t \]

7. Applied rewrites 17.1%

   \[\leadsto z \cdot \color{blue}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64
  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))