Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B

Percentage Accurate: 85.6% → 98.0%

Time: 3.8s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (a - 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, a)
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), intent (in) :: a
    code = x + ((y * (z - t)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (a - 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, a)
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), intent (in) :: a
    code = x + ((y * (z - t)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{z - t}{a - t}} + x \]

   8. sub-div N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-div N/A

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

   12. lower-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift--.f64 98.7

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

4. Applied rewrites 98.7%

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

Alternative 2: 54.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+55}:\\ \;\;\;\;y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) (- a t))))
   (if (<= t_1 -5e+55) y (if (<= t_1 2e+62) x y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / (a - t);
	double tmp;
	if (t_1 <= -5e+55) {
		tmp = y;
	} else if (t_1 <= 2e+62) {
		tmp = x;
	} else {
		tmp = y;
	}
	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, a)
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), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (z - t)) / (a - t)
    if (t_1 <= (-5d+55)) then
        tmp = y
    else if (t_1 <= 2d+62) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / (a - t);
	double tmp;
	if (t_1 <= -5e+55) {
		tmp = y;
	} else if (t_1 <= 2e+62) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / (a - t)
	tmp = 0
	if t_1 <= -5e+55:
		tmp = y
	elif t_1 <= 2e+62:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -5e+55)
		tmp = y;
	elseif (t_1 <= 2e+62)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / (a - t);
	tmp = 0.0;
	if (t_1 <= -5e+55)
		tmp = y;
	elseif (t_1 <= 2e+62)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+55], y, If[LessEqual[t$95$1, 2e+62], x, y]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < -5.00000000000000046e55 or 2.00000000000000007e62 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))

   1. Initial program 61.7%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 41.8

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \left(1 - \frac{z}{t}\right) \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 - \frac{z}{t}\right) \cdot y \]

      3. lower--.f64 N/A

         \[\leadsto \left(1 - \frac{z}{t}\right) \cdot y \]

      4. lower-/.f64 56.5

         \[\leadsto \left(1 - \frac{z}{t}\right) \cdot y \]

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in z around 0

      \[\leadsto y \]
   10. Step-by-step derivation

   11. Applied rewrites 29.4%

       \[\leadsto y \]

   if -5.00000000000000046e55 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < 2.00000000000000007e62

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 72.2%

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

Alternative 3: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+253}:\\ \;\;\;\;x + \mathsf{fma}\left(a, \frac{y}{t}, y\right)\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-126} \lor \neg \left(t \leq 1.5\right):\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.8e+253)
   (+ x (fma a (/ y t) y))
   (if (or (<= t -1.02e-126) (not (<= t 1.5)))
     (- x (* (- z t) (/ y t)))
     (fma y (/ (- z t) a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.8e+253) {
		tmp = x + fma(a, (y / t), y);
	} else if ((t <= -1.02e-126) || !(t <= 1.5)) {
		tmp = x - ((z - t) * (y / t));
	} else {
		tmp = fma(y, ((z - t) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.8e+253)
		tmp = Float64(x + fma(a, Float64(y / t), y));
	elseif ((t <= -1.02e-126) || !(t <= 1.5))
		tmp = Float64(x - Float64(Float64(z - t) * Float64(y / t)));
	else
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.8e+253], N[(x + N[(a * N[(y / t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -1.02e-126], N[Not[LessEqual[t, 1.5]], $MachinePrecision]], N[(x - N[(N[(z - t), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.8e253

   1. Initial program 55.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

         \[\leadsto x + \left(y + \frac{-1 \cdot \left(y \cdot z - a \cdot y\right)}{t}\right) \]

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

         \[\leadsto x + \left(\frac{y \cdot z - a \cdot y}{t} \cdot -1 + y\right) \]

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 98.3

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

   8. Applied rewrites 98.3%

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

   if -1.8e253 < t < -1.02000000000000004e-126 or 1.5 < t

   1. Initial program 80.4%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 69.7

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

   5. Applied rewrites 69.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 80.4

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

   7. Applied rewrites 80.4%

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

   if -1.02000000000000004e-126 < t < 1.5

   1. Initial program 91.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z - t}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+253}:\\ \;\;\;\;x + \mathsf{fma}\left(a, \frac{y}{t}, y\right)\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-126} \lor \neg \left(t \leq 1.5\right):\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-t}{a - t}, y, x\right)\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+98}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t) (- a t)) y x)))
   (if (<= t -1.02e-126)
     t_1
     (if (<= t 1.5)
       (fma y (/ (- z t) a) x)
       (if (<= t 9.5e+98) (- x (/ (* (- z t) y) t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-t / (a - t)), y, x);
	double tmp;
	if (t <= -1.02e-126) {
		tmp = t_1;
	} else if (t <= 1.5) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t <= 9.5e+98) {
		tmp = x - (((z - t) * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-t) / Float64(a - t)), y, x)
	tmp = 0.0
	if (t <= -1.02e-126)
		tmp = t_1;
	elseif (t <= 1.5)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t <= 9.5e+98)
		tmp = Float64(x - Float64(Float64(Float64(z - t) * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-t) / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -1.02e-126], t$95$1, If[LessEqual[t, 1.5], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 9.5e+98], N[(x - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.02000000000000004e-126 or 9.5000000000000001e98 < t

   1. Initial program 75.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z - t}{a - t}} + x \]

      8. sub-div N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(t\right)}{a - t}, y, x\right) \]

      2. lift-neg.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if -1.02000000000000004e-126 < t < 1.5

   1. Initial program 91.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z - t}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if 1.5 < t < 9.5000000000000001e98

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 90.5

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

   5. Applied rewrites 90.5%

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

Alternative 5: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+84} \lor \neg \left(t \leq 1.5\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.45e+84) (not (<= t 1.5))) (+ x y) (fma y (/ (- z t) a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.45e+84) || !(t <= 1.5)) {
		tmp = x + y;
	} else {
		tmp = fma(y, ((z - t) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.45e+84) || !(t <= 1.5))
		tmp = Float64(x + y);
	else
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.45e+84], N[Not[LessEqual[t, 1.5]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.44999999999999994e84 or 1.5 < t

   1. Initial program 75.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 81.6%

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

   if -1.44999999999999994e84 < t < 1.5

   1. Initial program 90.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z - t}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 81.3

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

   5. Applied rewrites 81.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+84} \lor \neg \left(t \leq 1.5\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{+94}:\\ \;\;\;\;x + \mathsf{fma}\left(a, \frac{y}{t}, y\right)\\ \mathbf{elif}\;t \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.25e+94)
   (+ x (fma a (/ y t) y))
   (if (<= t 1.5) (fma y (/ (- z t) a) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.25e+94) {
		tmp = x + fma(a, (y / t), y);
	} else if (t <= 1.5) {
		tmp = fma(y, ((z - t) / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.25e+94)
		tmp = Float64(x + fma(a, Float64(y / t), y));
	elseif (t <= 1.5)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.25e+94], N[(x + N[(a * N[(y / t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.24999999999999986e94

   1. Initial program 75.9%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

         \[\leadsto x + \left(y + \frac{-1 \cdot \left(y \cdot z - a \cdot y\right)}{t}\right) \]

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

         \[\leadsto x + \left(\frac{y \cdot z - a \cdot y}{t} \cdot -1 + y\right) \]

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-/.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if -2.24999999999999986e94 < t < 1.5

   1. Initial program 89.4%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z - t}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if 1.5 < t

   1. Initial program 76.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 78.1%

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

Alternative 7: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-126} \lor \neg \left(t \leq 1.25\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.02e-126) (not (<= t 1.25))) (+ x y) (fma y (/ z a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.02e-126) || !(t <= 1.25)) {
		tmp = x + y;
	} else {
		tmp = fma(y, (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.02e-126) || !(t <= 1.25))
		tmp = Float64(x + y);
	else
		tmp = fma(y, Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.02e-126], N[Not[LessEqual[t, 1.25]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.02000000000000004e-126 or 1.25 < t

   1. Initial program 78.4%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 73.8%

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

   if -1.02000000000000004e-126 < t < 1.25

   1. Initial program 91.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot z}{a} + \color{blue}{x} \]

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 87.1

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

   5. Applied rewrites 87.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-126} \lor \neg \left(t \leq 1.25\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-144}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-271}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.2e-144)
   (+ x y)
   (if (<= t 1.5e-271) (* y (/ z a)) (if (<= t 1.6e-8) x (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.2e-144) {
		tmp = x + y;
	} else if (t <= 1.5e-271) {
		tmp = y * (z / a);
	} else if (t <= 1.6e-8) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	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, a)
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), intent (in) :: a
    real(8) :: tmp
    if (t <= (-4.2d-144)) then
        tmp = x + y
    else if (t <= 1.5d-271) then
        tmp = y * (z / a)
    else if (t <= 1.6d-8) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.2e-144) {
		tmp = x + y;
	} else if (t <= 1.5e-271) {
		tmp = y * (z / a);
	} else if (t <= 1.6e-8) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -4.2e-144:
		tmp = x + y
	elif t <= 1.5e-271:
		tmp = y * (z / a)
	elif t <= 1.6e-8:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.2e-144)
		tmp = Float64(x + y);
	elseif (t <= 1.5e-271)
		tmp = Float64(y * Float64(z / a));
	elseif (t <= 1.6e-8)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -4.2e-144)
		tmp = x + y;
	elseif (t <= 1.5e-271)
		tmp = y * (z / a);
	elseif (t <= 1.6e-8)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.2e-144], N[(x + y), $MachinePrecision], If[LessEqual[t, 1.5e-271], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e-8], x, N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.2000000000000002e-144 or 1.6000000000000001e-8 < t

   1. Initial program 78.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 73.1%

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

   if -4.2000000000000002e-144 < t < 1.50000000000000001e-271

   1. Initial program 93.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{a - t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \frac{z}{\color{blue}{a - t}} \]

      4. lift--.f64 64.3

         \[\leadsto y \cdot \frac{z}{a - \color{blue}{t}} \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{y \cdot \frac{z}{a - t}} \]

   6. Taylor expanded in t around 0

      \[\leadsto y \cdot \frac{z}{a} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.8%

      \[\leadsto y \cdot \frac{z}{a} \]

   if 1.50000000000000001e-271 < t < 1.6000000000000001e-8

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 52.3%

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

Alternative 9: 61.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-144}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4e-144)
   (+ x y)
   (if (<= t 1.5e-271) (/ (* y z) a) (if (<= t 1.6e-8) x (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4e-144) {
		tmp = x + y;
	} else if (t <= 1.5e-271) {
		tmp = (y * z) / a;
	} else if (t <= 1.6e-8) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	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, a)
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), intent (in) :: a
    real(8) :: tmp
    if (t <= (-4d-144)) then
        tmp = x + y
    else if (t <= 1.5d-271) then
        tmp = (y * z) / a
    else if (t <= 1.6d-8) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4e-144) {
		tmp = x + y;
	} else if (t <= 1.5e-271) {
		tmp = (y * z) / a;
	} else if (t <= 1.6e-8) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -4e-144:
		tmp = x + y
	elif t <= 1.5e-271:
		tmp = (y * z) / a
	elif t <= 1.6e-8:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4e-144)
		tmp = Float64(x + y);
	elseif (t <= 1.5e-271)
		tmp = Float64(Float64(y * z) / a);
	elseif (t <= 1.6e-8)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -4e-144)
		tmp = x + y;
	elseif (t <= 1.5e-271)
		tmp = (y * z) / a;
	elseif (t <= 1.6e-8)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4e-144], N[(x + y), $MachinePrecision], If[LessEqual[t, 1.5e-271], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 1.6e-8], x, N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.9999999999999998e-144 or 1.6000000000000001e-8 < t

   1. Initial program 78.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 73.1%

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

   if -3.9999999999999998e-144 < t < 1.50000000000000001e-271

   1. Initial program 93.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{a - t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \frac{z}{\color{blue}{a - t}} \]

      4. lift--.f64 64.3

         \[\leadsto y \cdot \frac{z}{a - \color{blue}{t}} \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{y \cdot \frac{z}{a - t}} \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{a} \]

      2. lower-*.f64 57.3

         \[\leadsto \frac{y \cdot z}{a} \]

   8. Applied rewrites 57.3%

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]

   if 1.50000000000000001e-271 < t < 1.6000000000000001e-8

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 52.3%

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

Alternative 10: 63.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-121} \lor \neg \left(t \leq 1.6 \cdot 10^{-8}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3.3e-121) (not (<= t 1.6e-8))) (+ x y) x))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.3e-121) || !(t <= 1.6e-8)) {
		tmp = x + y;
	} else {
		tmp = x;
	}
	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, a)
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), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-3.3d-121)) .or. (.not. (t <= 1.6d-8))) then
        tmp = x + y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.3e-121) || !(t <= 1.6e-8)) {
		tmp = x + y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -3.3e-121) or not (t <= 1.6e-8):
		tmp = x + y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3.3e-121) || !(t <= 1.6e-8))
		tmp = Float64(x + y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -3.3e-121) || ~((t <= 1.6e-8)))
		tmp = x + y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3.3e-121], N[Not[LessEqual[t, 1.6e-8]], $MachinePrecision]], N[(x + y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < -3.3000000000000001e-121 or 1.6000000000000001e-8 < t

   1. Initial program 77.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 72.8%

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

   if -3.3000000000000001e-121 < t < 1.6000000000000001e-8

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 45.3%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-121} \lor \neg \left(t \leq 1.6 \cdot 10^{-8}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.5% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 46.6%

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

Reproduce

herbie shell --seed 2025085 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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