SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.5% → 97.0%

Time: 9.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 93.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 96.4

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

4. Applied rewrites 96.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-tanh.f64 N/A

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

   5. tanh-def-a N/A

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

   6. cosh-undef N/A

      \[\leadsto \mathsf{fma}\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{e^{\frac{x}{y}} - e^{\mathsf{neg}\left(\frac{x}{y}\right)}}{\color{blue}{2 \cdot \cosh \left(\frac{x}{y}\right)}}\right), y, x\right) \]

   7. lift-cosh.f64 N/A

      \[\leadsto \mathsf{fma}\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{e^{\frac{x}{y}} - e^{\mathsf{neg}\left(\frac{x}{y}\right)}}{2 \cdot \color{blue}{\cosh \left(\frac{x}{y}\right)}}\right), y, x\right) \]

   8. associate-/l/ N/A

      \[\leadsto \mathsf{fma}\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{\frac{e^{\frac{x}{y}} - e^{\mathsf{neg}\left(\frac{x}{y}\right)}}{2}}{\cosh \left(\frac{x}{y}\right)}}\right), y, x\right) \]

   9. sinh-def N/A

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

   10. lift-sinh.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. *-lft-identity N/A

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

   13. lift-*.f64 N/A

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

   14. flip3-- N/A

       \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{\frac{{\tanh \left(\frac{t}{y}\right)}^{3} - {\left(1 \cdot \frac{\sinh \left(\frac{x}{y}\right)}{\cosh \left(\frac{x}{y}\right)}\right)}^{3}}{\tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{t}{y}\right) + \left(\left(1 \cdot \frac{\sinh \left(\frac{x}{y}\right)}{\cosh \left(\frac{x}{y}\right)}\right) \cdot \left(1 \cdot \frac{\sinh \left(\frac{x}{y}\right)}{\cosh \left(\frac{x}{y}\right)}\right) + \tanh \left(\frac{t}{y}\right) \cdot \left(1 \cdot \frac{\sinh \left(\frac{x}{y}\right)}{\cosh \left(\frac{x}{y}\right)}\right)\right)}}, y, x\right) \]

6. Applied rewrites 96.4%

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

7. Final simplification 96.4%

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

Alternative 2: 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 90.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 96.4

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

4. Applied rewrites 96.4%

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

Alternative 3: 84.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0}{y} \cdot z, y, x\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{\frac{x}{y} \cdot 2}{\mathsf{fma}\left(\frac{x}{y}, -2, 1\right) + 1}\right) \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right), z \cdot y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -7e+98)
   (fma (* (/ 0.0 y) z) y x)
   (if (<= x 2.4e+77)
     (fma
      (*
       (- (tanh (/ t y)) (/ (* (/ x y) 2.0) (+ (fma (/ x y) -2.0 1.0) 1.0)))
       z)
      y
      x)
     (fma (- (/ t y) (tanh (/ x y))) (* z y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7e+98) {
		tmp = fma(((0.0 / y) * z), y, x);
	} else if (x <= 2.4e+77) {
		tmp = fma(((tanh((t / y)) - (((x / y) * 2.0) / (fma((x / y), -2.0, 1.0) + 1.0))) * z), y, x);
	} else {
		tmp = fma(((t / y) - tanh((x / y))), (z * y), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7e+98)
		tmp = fma(Float64(Float64(0.0 / y) * z), y, x);
	elseif (x <= 2.4e+77)
		tmp = fma(Float64(Float64(tanh(Float64(t / y)) - Float64(Float64(Float64(x / y) * 2.0) / Float64(fma(Float64(x / y), -2.0, 1.0) + 1.0))) * z), y, x);
	else
		tmp = fma(Float64(Float64(t / y) - tanh(Float64(x / y))), Float64(z * y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -7e+98], N[(N[(N[(0.0 / y), $MachinePrecision] * z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[x, 2.4e+77], N[(N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(x / y), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(x / y), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7e98

   1. Initial program 93.8%

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

      1. lift-tanh.f64 N/A

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

      2. tanh-def-c N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - e^{-2 \cdot \frac{x}{y}}}{1 + e^{-2 \cdot \frac{x}{y}}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - e^{-2 \cdot \frac{x}{y}}}{1 + e^{-2 \cdot \frac{x}{y}}}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{\color{blue}{1 - e^{-2 \cdot \frac{x}{y}}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      5. exp-prod N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - \color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - \color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      7. lower-exp.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\color{blue}{\left(e^{-2}\right)}}^{\left(\frac{x}{y}\right)}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      8. +-commutative N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{e^{-2 \cdot \frac{x}{y}} + 1}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{e^{-2 \cdot \frac{x}{y}} + 1}}\right) \]

      10. exp-prod N/A

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}} + 1}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}} + 1}\right) \]

      12. lower-exp.f64 63.8

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{{\color{blue}{\left(e^{-2}\right)}}^{\left(\frac{x}{y}\right)} + 1}\right) \]

   4. Applied rewrites 63.8%

      \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)} + 1}}\right) \]

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\frac{\left(t + \left(-1 \cdot x + \left(\frac{-1}{2} \cdot \frac{{x}^{2}}{y} + \left(\frac{1}{2} \cdot \frac{-1 \cdot {t}^{2} + {t}^{2}}{y} + \frac{{x}^{2}}{y}\right)\right)\right)\right) - -1 \cdot \frac{x \cdot \left(-1 \cdot x - \frac{-1}{2} \cdot x\right)}{y}}{y}} \]

   7. Applied rewrites 13.0%

      \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\frac{\left(\mathsf{fma}\left(-1, x, t\right) + \frac{\mathsf{fma}\left(-0.5, x \cdot x, \mathsf{fma}\left(0 \cdot \left(t \cdot t\right), 0.5, x \cdot x\right)\right)}{y}\right) - \left(-x\right) \cdot \frac{-0.5 \cdot x}{y}}{y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(y \cdot z\right) \cdot \frac{\frac{\left(\frac{-1}{2} \cdot {x}^{2} + {x}^{2}\right) - \frac{1}{2} \cdot {x}^{2}}{y}}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

       \[\leadsto x + \left(y \cdot z\right) \cdot \frac{\frac{\left(x \cdot x\right) \cdot 0}{y}}{y} \]

   12. Applied rewrites 91.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0}{y} \cdot z, y, x\right)} \]

   if -7e98 < x < 2.3999999999999999e77

   1. Initial program 87.5%

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

      1. lift-tanh.f64 N/A

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

      2. tanh-def-c N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - e^{-2 \cdot \frac{x}{y}}}{1 + e^{-2 \cdot \frac{x}{y}}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - e^{-2 \cdot \frac{x}{y}}}{1 + e^{-2 \cdot \frac{x}{y}}}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{\color{blue}{1 - e^{-2 \cdot \frac{x}{y}}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      5. exp-prod N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - \color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - \color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      7. lower-exp.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\color{blue}{\left(e^{-2}\right)}}^{\left(\frac{x}{y}\right)}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      8. +-commutative N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{e^{-2 \cdot \frac{x}{y}} + 1}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{e^{-2 \cdot \frac{x}{y}} + 1}}\right) \]

      10. exp-prod N/A

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}} + 1}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}} + 1}\right) \]

      12. lower-exp.f64 67.1

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{{\color{blue}{\left(e^{-2}\right)}}^{\left(\frac{x}{y}\right)} + 1}\right) \]

   4. Applied rewrites 67.1%

      \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)} + 1}}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{\left(1 + -2 \cdot \frac{x}{y}\right)} + 1}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{\left(-2 \cdot \frac{x}{y} + 1\right)} + 1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right)} + 1}\right) \]

      3. lower-/.f64 60.5

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\mathsf{fma}\left(-2, \color{blue}{\frac{x}{y}}, 1\right) + 1}\right) \]

   7. Applied rewrites 60.5%

      \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right)} + 1}\right) \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{\color{blue}{2 \cdot \frac{x}{y}}}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right) + 1}\right) \]

      2. lower-/.f64 78.0

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{2 \cdot \color{blue}{\frac{x}{y}}}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right) + 1}\right) \]

   10. Applied rewrites 78.0%

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

       1. lift-+.f64 N/A

          \[\leadsto \color{blue}{x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{2 \cdot \frac{x}{y}}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right) + 1}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{2 \cdot \frac{x}{y}}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right) + 1}\right) + x} \]

       3. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{2 \cdot \frac{x}{y}}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right) + 1}\right)} + x \]

       4. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot z\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{2 \cdot \frac{x}{y}}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right) + 1}\right) + x \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{2 \cdot \frac{x}{y}}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right) + 1}\right)\right)} + x \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{2 \cdot \frac{x}{y}}{\mathsf{fma}\left(-2, \frac{x}{y}, 1\right) + 1}\right)\right) \cdot y} + x \]

   12. Applied rewrites 84.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{\frac{x}{y} \cdot 2}{\mathsf{fma}\left(\frac{x}{y}, -2, 1\right) + 1}\right) \cdot z, y, x\right)} \]

   if 2.3999999999999999e77 < x

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 88.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.2

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

   7. Applied rewrites 88.2%

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

Alternative 4: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+53} \lor \neg \left(y \leq 1.32 \cdot 10^{-62}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right) \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0}{y} \cdot z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -6.8e+53) (not (<= y 1.32e-62)))
   (fma (* (- (tanh (/ t y)) (/ x y)) y) z x)
   (fma (* (/ 0.0 y) z) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6.8e+53) || !(y <= 1.32e-62)) {
		tmp = fma(((tanh((t / y)) - (x / y)) * y), z, x);
	} else {
		tmp = fma(((0.0 / y) * z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -6.8e+53) || !(y <= 1.32e-62))
		tmp = fma(Float64(Float64(tanh(Float64(t / y)) - Float64(x / y)) * y), z, x);
	else
		tmp = fma(Float64(Float64(0.0 / y) * z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -6.8e+53], N[Not[LessEqual[y, 1.32e-62]], $MachinePrecision]], N[(N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(0.0 / y), $MachinePrecision] * z), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.79999999999999995e53 or 1.31999999999999997e-62 < y

   1. Initial program 81.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 73.7

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

   5. Applied rewrites 73.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

   if -6.79999999999999995e53 < y < 1.31999999999999997e-62

   1. Initial program 99.2%

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

      1. lift-tanh.f64 N/A

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

      2. tanh-def-c N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - e^{-2 \cdot \frac{x}{y}}}{1 + e^{-2 \cdot \frac{x}{y}}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - e^{-2 \cdot \frac{x}{y}}}{1 + e^{-2 \cdot \frac{x}{y}}}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{\color{blue}{1 - e^{-2 \cdot \frac{x}{y}}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      5. exp-prod N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - \color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - \color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      7. lower-exp.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\color{blue}{\left(e^{-2}\right)}}^{\left(\frac{x}{y}\right)}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      8. +-commutative N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{e^{-2 \cdot \frac{x}{y}} + 1}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{e^{-2 \cdot \frac{x}{y}} + 1}}\right) \]

      10. exp-prod N/A

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}} + 1}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}} + 1}\right) \]

      12. lower-exp.f64 65.4

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{{\color{blue}{\left(e^{-2}\right)}}^{\left(\frac{x}{y}\right)} + 1}\right) \]

   4. Applied rewrites 65.4%

      \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)} + 1}}\right) \]

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\frac{\left(t + \left(-1 \cdot x + \left(\frac{-1}{2} \cdot \frac{{x}^{2}}{y} + \left(\frac{1}{2} \cdot \frac{-1 \cdot {t}^{2} + {t}^{2}}{y} + \frac{{x}^{2}}{y}\right)\right)\right)\right) - -1 \cdot \frac{x \cdot \left(-1 \cdot x - \frac{-1}{2} \cdot x\right)}{y}}{y}} \]

   7. Applied rewrites 23.3%

      \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\frac{\left(\mathsf{fma}\left(-1, x, t\right) + \frac{\mathsf{fma}\left(-0.5, x \cdot x, \mathsf{fma}\left(0 \cdot \left(t \cdot t\right), 0.5, x \cdot x\right)\right)}{y}\right) - \left(-x\right) \cdot \frac{-0.5 \cdot x}{y}}{y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(y \cdot z\right) \cdot \frac{\frac{\left(\frac{-1}{2} \cdot {x}^{2} + {x}^{2}\right) - \frac{1}{2} \cdot {x}^{2}}{y}}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.5%

       \[\leadsto x + \left(y \cdot z\right) \cdot \frac{\frac{\left(x \cdot x\right) \cdot 0}{y}}{y} \]

   12. Applied rewrites 79.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0}{y} \cdot z, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+53} \lor \neg \left(y \leq 1.32 \cdot 10^{-62}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right) \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0}{y} \cdot z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+87} \lor \neg \left(y \leq 2.9 \cdot 10^{+61}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0}{y} \cdot z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.52e+87) (not (<= y 2.9e+61)))
   (fma (- t x) z x)
   (fma (* (/ 0.0 y) z) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.52e+87) || !(y <= 2.9e+61)) {
		tmp = fma((t - x), z, x);
	} else {
		tmp = fma(((0.0 / y) * z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.52e+87) || !(y <= 2.9e+61))
		tmp = fma(Float64(t - x), z, x);
	else
		tmp = fma(Float64(Float64(0.0 / y) * z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.52e+87], N[Not[LessEqual[y, 2.9e+61]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(0.0 / y), $MachinePrecision] * z), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.51999999999999991e87 or 2.9000000000000001e61 < y

   1. Initial program 77.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if -1.51999999999999991e87 < y < 2.9000000000000001e61

   1. Initial program 98.2%

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

      1. lift-tanh.f64 N/A

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

      2. tanh-def-c N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - e^{-2 \cdot \frac{x}{y}}}{1 + e^{-2 \cdot \frac{x}{y}}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - e^{-2 \cdot \frac{x}{y}}}{1 + e^{-2 \cdot \frac{x}{y}}}}\right) \]

      4. lower--.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{\color{blue}{1 - e^{-2 \cdot \frac{x}{y}}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      5. exp-prod N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - \color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - \color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      7. lower-exp.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\color{blue}{\left(e^{-2}\right)}}^{\left(\frac{x}{y}\right)}}{1 + e^{-2 \cdot \frac{x}{y}}}\right) \]

      8. +-commutative N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{e^{-2 \cdot \frac{x}{y}} + 1}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{e^{-2 \cdot \frac{x}{y}} + 1}}\right) \]

      10. exp-prod N/A

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}} + 1}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{\color{blue}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}} + 1}\right) \]

      12. lower-exp.f64 64.6

          \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{{\color{blue}{\left(e^{-2}\right)}}^{\left(\frac{x}{y}\right)} + 1}\right) \]

   4. Applied rewrites 64.6%

      \[\leadsto x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \color{blue}{\frac{1 - {\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)}}{{\left(e^{-2}\right)}^{\left(\frac{x}{y}\right)} + 1}}\right) \]

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\frac{\left(t + \left(-1 \cdot x + \left(\frac{-1}{2} \cdot \frac{{x}^{2}}{y} + \left(\frac{1}{2} \cdot \frac{-1 \cdot {t}^{2} + {t}^{2}}{y} + \frac{{x}^{2}}{y}\right)\right)\right)\right) - -1 \cdot \frac{x \cdot \left(-1 \cdot x - \frac{-1}{2} \cdot x\right)}{y}}{y}} \]

   7. Applied rewrites 25.7%

      \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\frac{\left(\mathsf{fma}\left(-1, x, t\right) + \frac{\mathsf{fma}\left(-0.5, x \cdot x, \mathsf{fma}\left(0 \cdot \left(t \cdot t\right), 0.5, x \cdot x\right)\right)}{y}\right) - \left(-x\right) \cdot \frac{-0.5 \cdot x}{y}}{y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(y \cdot z\right) \cdot \frac{\frac{\left(\frac{-1}{2} \cdot {x}^{2} + {x}^{2}\right) - \frac{1}{2} \cdot {x}^{2}}{y}}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.5%

       \[\leadsto x + \left(y \cdot z\right) \cdot \frac{\frac{\left(x \cdot x\right) \cdot 0}{y}}{y} \]

   12. Applied rewrites 75.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0}{y} \cdot z, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+87} \lor \neg \left(y \leq 2.9 \cdot 10^{+61}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0}{y} \cdot z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{if}\;y \leq -6.9 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y} \cdot z, y, x\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- t x) z x)))
   (if (<= y -6.9e+53)
     t_1
     (if (<= y -4.8e-121)
       (fma (* (/ t y) z) y x)
       (if (<= y 7.2e-109) (fma (- x) z 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 <= -6.9e+53) {
		tmp = t_1;
	} else if (y <= -4.8e-121) {
		tmp = fma(((t / y) * z), y, x);
	} else if (y <= 7.2e-109) {
		tmp = fma(-x, z, 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 <= -6.9e+53)
		tmp = t_1;
	elseif (y <= -4.8e-121)
		tmp = fma(Float64(Float64(t / y) * z), y, x);
	elseif (y <= 7.2e-109)
		tmp = fma(Float64(-x), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[y, -6.9e+53], t$95$1, If[LessEqual[y, -4.8e-121], N[(N[(N[(t / y), $MachinePrecision] * z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 7.2e-109], N[((-x) * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.9000000000000002e53 or 7.2000000000000001e-109 < y

   1. Initial program 82.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 73.9

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

   5. Applied rewrites 73.9%

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

   if -6.9000000000000002e53 < y < -4.80000000000000007e-121

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 47.0

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

   7. Applied rewrites 47.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.7%

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

   if -4.80000000000000007e-121 < y < 7.2000000000000001e-109

   1. Initial program 98.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 34.8

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.1%

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

Alternative 7: 64.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-121} \lor \neg \left(y \leq 7.2 \cdot 10^{-109}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -7e-121) (not (<= y 7.2e-109)))
   (fma (- t x) z x)
   (fma (- x) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -7e-121) || !(y <= 7.2e-109)) {
		tmp = fma((t - x), z, x);
	} else {
		tmp = fma(-x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -7e-121) || !(y <= 7.2e-109))
		tmp = fma(Float64(t - x), z, x);
	else
		tmp = fma(Float64(-x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -7e-121], N[Not[LessEqual[y, 7.2e-109]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision], N[((-x) * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999985e-121 or 7.2000000000000001e-109 < y

   1. Initial program 86.4%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if -6.99999999999999985e-121 < y < 7.2000000000000001e-109

   1. Initial program 98.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 34.8

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-121} \lor \neg \left(y \leq 7.2 \cdot 10^{-109}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7200 \lor \neg \left(z \leq 2600000000\right):\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7200.0) (not (<= z 2600000000.0)))
   (* (- t x) z)
   (fma (- x) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7200.0) || !(z <= 2600000000.0)) {
		tmp = (t - x) * z;
	} else {
		tmp = fma(-x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7200.0) || !(z <= 2600000000.0))
		tmp = Float64(Float64(t - x) * z);
	else
		tmp = fma(Float64(-x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -7200.0], N[Not[LessEqual[z, 2600000000.0]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision], N[((-x) * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7200 or 2.6e9 < z

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 36.0

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

   5. Applied rewrites 36.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 18.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 36.0%

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

   if -7200 < z < 2.6e9

   1. Initial program 98.4%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.9%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7200 \lor \neg \left(z \leq 2600000000\right):\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 20.6% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-22} \lor \neg \left(x \leq 2.8 \cdot 10^{-121}\right):\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.9e-22) (not (<= x 2.8e-121))) (* (- x) z) (* z t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.9e-22) || !(x <= 2.8e-121)) {
		tmp = -x * z;
	} else {
		tmp = z * t;
	}
	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) :: tmp
    if ((x <= (-3.9d-22)) .or. (.not. (x <= 2.8d-121))) then
        tmp = -x * z
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.9e-22) || !(x <= 2.8e-121)) {
		tmp = -x * z;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.9e-22) or not (x <= 2.8e-121):
		tmp = -x * z
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.9e-22) || !(x <= 2.8e-121))
		tmp = Float64(Float64(-x) * z);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.9e-22) || ~((x <= 2.8e-121)))
		tmp = -x * z;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.9e-22], N[Not[LessEqual[x, 2.8e-121]], $MachinePrecision]], N[((-x) * z), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.89999999999999998e-22 or 2.8000000000000001e-121 < 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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 17.2%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 14.1%

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

   if -3.89999999999999998e-22 < x < 2.8000000000000001e-121

   1. Initial program 85.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.4%

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

4. Final simplification 18.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-22} \lor \neg \left(x \leq 2.8 \cdot 10^{-121}\right):\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.4% accurate, 26.6× speedup

Math

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

FPCore

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

C

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

Fortran

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 = (t - x) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 57.5

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

5. Applied rewrites 57.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 13.4%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 22.3%

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

Alternative 11: 17.0% accurate, 39.8× speedup

Math

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

FPCore

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

C

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

Fortran

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 90.5%

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

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 57.5

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

5. Applied rewrites 57.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 13.4%

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

Developer Target 1: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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(y * Float64(z * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))))))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024352 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y)))))))

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