SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.2% → 98.0%

Time: 8.1s

Alternatives: 10

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: 93.2% 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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	return fma(Float64(Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))) * y), z, 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] * y), $MachinePrecision] * z + x), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 98.1

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

4. Applied rewrites 98.1%

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

Alternative 2: 76.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right) \cdot y, z, x\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -42000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t \cdot t}{x + t}, z, x\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right), z \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (* (- (tanh (/ t y)) (/ x y)) y) z x)))
   (if (<= t -4.5e+44)
     t_1
     (if (<= t -42000000000000.0)
       (fma (/ (* t t) (+ x t)) z x)
       (if (<= t 3.8e-41) (fma (- (/ t y) (tanh (/ x y))) (* z y) x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(((tanh((t / y)) - (x / y)) * y), z, x);
	double tmp;
	if (t <= -4.5e+44) {
		tmp = t_1;
	} else if (t <= -42000000000000.0) {
		tmp = fma(((t * t) / (x + t)), z, x);
	} else if (t <= 3.8e-41) {
		tmp = fma(((t / y) - tanh((x / y))), (z * y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(Float64(tanh(Float64(t / y)) - Float64(x / y)) * y), z, x)
	tmp = 0.0
	if (t <= -4.5e+44)
		tmp = t_1;
	elseif (t <= -42000000000000.0)
		tmp = fma(Float64(Float64(t * t) / Float64(x + t)), z, x);
	elseif (t <= 3.8e-41)
		tmp = fma(Float64(Float64(t / y) - tanh(Float64(x / y))), Float64(z * y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t, -4.5e+44], t$95$1, If[LessEqual[t, -42000000000000.0], N[(N[(N[(t * t), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t, 3.8e-41], N[(N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.5e44 or 3.79999999999999979e-41 < t

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if -4.5e44 < t < -4.2e13

   1. Initial program 91.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-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 55.9

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

   5. Applied rewrites 55.9%

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 91.1%

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

   if -4.2e13 < t < 3.79999999999999979e-41

   1. Initial program 91.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{t - \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot {t}^{2}\right)}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{t - \frac{\frac{-1}{2} \cdot \color{blue}{0}}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right) \]

      8. metadata-eval N/A

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

      9. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{t - \frac{\color{blue}{0 \cdot {x}^{2}}}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right) \]

      10. metadata-eval N/A

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

      11. distribute-lft1-in N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{t - \frac{\color{blue}{0} \cdot {x}^{2}}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right) \]

      16. mul0-lft 91.8

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

   7. Applied rewrites 91.8%

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{t - \frac{0}{y}}{y}} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{t - \frac{0}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y\right) \cdot z + x} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{t - \frac{0}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y\right)} \cdot z + x \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{t - \frac{0}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)} + x \]

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

   9. Applied rewrites 87.7%

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

Alternative 3: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+35} \lor \neg \left(t \leq 4 \cdot 10^{-41}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right) \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.2e+35) (not (<= t 4e-41)))
   (fma (* (- (tanh (/ t y)) (/ x y)) y) z x)
   (fma (* (- (/ t y) (tanh (/ x y))) y) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.2e+35) || !(t <= 4e-41)) {
		tmp = fma(((tanh((t / y)) - (x / y)) * y), z, x);
	} else {
		tmp = fma((((t / y) - tanh((x / y))) * y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.2e+35) || !(t <= 4e-41))
		tmp = fma(Float64(Float64(tanh(Float64(t / y)) - Float64(x / y)) * y), z, x);
	else
		tmp = fma(Float64(Float64(Float64(t / y) - tanh(Float64(x / y))) * y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.2e+35], N[Not[LessEqual[t, 4e-41]], $MachinePrecision]], N[(N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.1999999999999998e35 or 4.00000000000000002e-41 < t

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 65.7

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

   7. Applied rewrites 65.7%

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

   if -4.1999999999999998e35 < t < 4.00000000000000002e-41

   1. Initial program 91.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{t - \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot {t}^{2}\right)}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{t - \frac{\frac{-1}{2} \cdot \color{blue}{0}}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right) \]

      8. metadata-eval N/A

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

      9. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{t - \frac{\color{blue}{0 \cdot {x}^{2}}}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right) \]

      10. metadata-eval N/A

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

      11. distribute-lft1-in N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{t - \frac{\color{blue}{0} \cdot {x}^{2}}{y}}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right) \]

      16. mul0-lft 90.7

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

   7. Applied rewrites 90.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+35} \lor \neg \left(t \leq 4 \cdot 10^{-41}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right) \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t \cdot t}{x + t}, z, x\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right) \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t \cdot t}{\left(x - t\right) \cdot \left(x + t\right)} \cdot \left(x - t\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.05e+140)
   (fma (/ (* t t) (+ x t)) z x)
   (if (<= x 5e+125)
     (fma (* (- (tanh (/ t y)) (/ x y)) y) z x)
     (fma (* (/ (* t t) (* (- x t) (+ x t))) (- x t)) z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e+140) {
		tmp = fma(((t * t) / (x + t)), z, x);
	} else if (x <= 5e+125) {
		tmp = fma(((tanh((t / y)) - (x / y)) * y), z, x);
	} else {
		tmp = fma((((t * t) / ((x - t) * (x + t))) * (x - t)), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.05e+140)
		tmp = fma(Float64(Float64(t * t) / Float64(x + t)), z, x);
	elseif (x <= 5e+125)
		tmp = fma(Float64(Float64(tanh(Float64(t / y)) - Float64(x / y)) * y), z, x);
	else
		tmp = fma(Float64(Float64(Float64(t * t) / Float64(Float64(x - t) * Float64(x + t))) * Float64(x - t)), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.05e+140], N[(N[(N[(t * t), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[x, 5e+125], N[(N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(N[(t * t), $MachinePrecision] / N[(N[(x - t), $MachinePrecision] * N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0500000000000001e140

   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. 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 48.7

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

   5. Applied rewrites 48.7%

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.1%

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

   if -1.0500000000000001e140 < x < 4.99999999999999962e125

   1. Initial program 92.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 76.7

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

   7. Applied rewrites 76.7%

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

   if 4.99999999999999962e125 < x

   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. 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 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 14.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.8%

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

   12. Applied rewrites 65.2%

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

Alternative 5: 60.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.92 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t \cdot t}{x + t}, z, x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \frac{t}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.92e+28)
   (fma (/ (* t t) (+ x t)) z x)
   (if (<= z 1.12e-144)
     (fma (- x) z x)
     (if (<= z 5.8e+78) (fma (* z y) (/ t y) x) (fma (- t x) z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.92e+28) {
		tmp = fma(((t * t) / (x + t)), z, x);
	} else if (z <= 1.12e-144) {
		tmp = fma(-x, z, x);
	} else if (z <= 5.8e+78) {
		tmp = fma((z * y), (t / y), x);
	} else {
		tmp = fma((t - x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.92e+28)
		tmp = fma(Float64(Float64(t * t) / Float64(x + t)), z, x);
	elseif (z <= 1.12e-144)
		tmp = fma(Float64(-x), z, x);
	elseif (z <= 5.8e+78)
		tmp = fma(Float64(z * y), Float64(t / y), x);
	else
		tmp = fma(Float64(t - x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.92e+28], N[(N[(N[(t * t), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 1.12e-144], N[((-x) * z + x), $MachinePrecision], If[LessEqual[z, 5.8e+78], N[(N[(z * y), $MachinePrecision] * N[(t / y), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.91999999999999998e28

   1. Initial program 93.5%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 33.9

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

   5. Applied rewrites 33.9%

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

   7. Applied rewrites 28.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 38.0%

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

   if -1.91999999999999998e28 < z < 1.12e-144

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-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 75.8

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

   5. Applied rewrites 75.8%

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

   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 90.3%

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

   if 1.12e-144 < z < 5.80000000000000034e78

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 43.6

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

   5. Applied rewrites 43.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 43.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 43.6

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

   7. Applied rewrites 43.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.9%

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

   if 5.80000000000000034e78 < z

   1. Initial program 84.0%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 42.9

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

   5. Applied rewrites 42.9%

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

Alternative 6: 62.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \frac{t}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- t x) z x)))
   (if (<= z -1.75e-12)
     t_1
     (if (<= z 1.12e-144)
       (fma (- x) z x)
       (if (<= z 5.8e+78) (fma (* z y) (/ t y) 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 (z <= -1.75e-12) {
		tmp = t_1;
	} else if (z <= 1.12e-144) {
		tmp = fma(-x, z, x);
	} else if (z <= 5.8e+78) {
		tmp = fma((z * y), (t / y), 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 (z <= -1.75e-12)
		tmp = t_1;
	elseif (z <= 1.12e-144)
		tmp = fma(Float64(-x), z, x);
	elseif (z <= 5.8e+78)
		tmp = fma(Float64(z * y), Float64(t / y), 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[z, -1.75e-12], t$95$1, If[LessEqual[z, 1.12e-144], N[((-x) * z + x), $MachinePrecision], If[LessEqual[z, 5.8e+78], N[(N[(z * y), $MachinePrecision] * N[(t / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.75e-12 or 5.80000000000000034e78 < z

   1. Initial program 88.7%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-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 39.9

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

   5. Applied rewrites 39.9%

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

   if -1.75e-12 < z < 1.12e-144

   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. 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 75.6

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

   5. Applied rewrites 75.6%

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

   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 92.6%

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

   if 1.12e-144 < z < 5.80000000000000034e78

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 43.6

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

   5. Applied rewrites 43.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 43.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 43.6

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

   7. Applied rewrites 43.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.9%

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

Alternative 7: 62.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-12} \lor \neg \left(z \leq 4 \cdot 10^{-146}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.75e-12) (not (<= z 4e-146)))
   (fma (- t x) z x)
   (fma (- x) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.75e-12) || !(z <= 4e-146)) {
		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 ((z <= -1.75e-12) || !(z <= 4e-146))
		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[z, -1.75e-12], N[Not[LessEqual[z, 4e-146]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * z + x), $MachinePrecision], N[((-x) * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.75e-12 or 4.0000000000000001e-146 < z

   1. Initial program 91.5%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 42.2

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

   5. Applied rewrites 42.2%

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

   if -1.75e-12 < z < 4.0000000000000001e-146

   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. 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 75.6

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

   5. Applied rewrites 75.6%

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

   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 92.6%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-12} \lor \neg \left(z \leq 4 \cdot 10^{-146}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-10} \lor \neg \left(z \leq 5.8 \cdot 10^{-10}\right):\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.3e-10) (not (<= z 5.8e-10))) (* (- t x) z) (fma (- x) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.3e-10) || !(z <= 5.8e-10)) {
		tmp = (t - x) * z;
	} else {
		tmp = fma(-x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.3e-10) || !(z <= 5.8e-10))
		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, -3.3e-10], N[Not[LessEqual[z, 5.8e-10]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision], N[((-x) * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e-10 or 5.79999999999999962e-10 < z

   1. Initial program 90.2%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   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 37.6

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

   5. Applied rewrites 37.6%

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.5%

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

   11. Taylor expanded in z around inf

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

   13. Applied rewrites 36.9%

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

   if -3.3e-10 < z < 5.79999999999999962e-10

   1. Initial program 99.9%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   2. 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 75.0

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

   5. Applied rewrites 75.0%

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

   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 87.4%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-10} \lor \neg \left(z \leq 5.8 \cdot 10^{-10}\right):\\ \;\;\;\;\left(t - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.6% 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 94.8%

   \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
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 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 39.6%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 51.1%

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

11. Taylor expanded in z around inf

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

13. Applied rewrites 23.5%

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

14. Final simplification 23.5%

    \[\leadsto \left(t - x\right) \cdot z \]
15. Add Preprocessing

Alternative 10: 16.8% 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 94.8%

   \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
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 55.0

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

5. Applied rewrites 55.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 15.4%

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

Developer Target 1: 97.3% 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 2024363 
(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))))))