Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A

Percentage Accurate: 98.2% → 98.2%

Time: 3.9s

Alternatives: 16

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

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) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift--.f64 97.8

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

4. Applied rewrites 97.8%

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

Alternative 2: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{-t}{z - a}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{-a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ (- t) (- z a)) y x)))
   (if (<= t_1 -2e+17)
     t_2
     (if (<= t_1 1e-41)
       (fma (/ (- z t) (- a)) y x)
       (if (<= t_1 2.0) (fma y (/ z (- z a)) x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((-t / (z - a)), y, x);
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = t_2;
	} else if (t_1 <= 1e-41) {
		tmp = fma(((z - t) / -a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(Float64(-t) / Float64(z - a)), y, x)
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = t_2;
	elseif (t_1 <= 1e-41)
		tmp = fma(Float64(Float64(z - t) / Float64(-a)), y, x);
	elseif (t_1 <= 2.0)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-t) / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], t$95$2, If[LessEqual[t$95$1, 1e-41], N[(N[(N[(z - t), $MachinePrecision] / (-a)), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e17 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if -2e17 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000001e-41

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 1.00000000000000001e-41 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 71.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+120}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+204}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -4e+120)
     (* (/ y a) t)
     (if (<= t_1 2e-61) x (if (<= t_1 5e+204) (+ x y) (/ (* y t) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -4e+120) {
		tmp = (y / a) * t;
	} else if (t_1 <= 2e-61) {
		tmp = x;
	} else if (t_1 <= 5e+204) {
		tmp = x + y;
	} else {
		tmp = (y * t) / a;
	}
	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 = (z - t) / (z - a)
    if (t_1 <= (-4d+120)) then
        tmp = (y / a) * t
    else if (t_1 <= 2d-61) then
        tmp = x
    else if (t_1 <= 5d+204) then
        tmp = x + y
    else
        tmp = (y * t) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -4e+120) {
		tmp = (y / a) * t;
	} else if (t_1 <= 2e-61) {
		tmp = x;
	} else if (t_1 <= 5e+204) {
		tmp = x + y;
	} else {
		tmp = (y * t) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if t_1 <= -4e+120:
		tmp = (y / a) * t
	elif t_1 <= 2e-61:
		tmp = x
	elif t_1 <= 5e+204:
		tmp = x + y
	else:
		tmp = (y * t) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -4e+120)
		tmp = Float64(Float64(y / a) * t);
	elseif (t_1 <= 2e-61)
		tmp = x;
	elseif (t_1 <= 5e+204)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(y * t) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	tmp = 0.0;
	if (t_1 <= -4e+120)
		tmp = (y / a) * t;
	elseif (t_1 <= 2e-61)
		tmp = x;
	elseif (t_1 <= 5e+204)
		tmp = x + y;
	else
		tmp = (y * t) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+120], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e-61], x, If[LessEqual[t$95$1, 5e+204], N[(x + y), $MachinePrecision], N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.9999999999999999e120

   1. Initial program 95.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 63.0

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

   8. Applied rewrites 63.0%

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

   if -3.9999999999999999e120 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-61

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 72.6%

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

   if 2.0000000000000001e-61 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000008e204

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 84.2%

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

   if 5.00000000000000008e204 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 79.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 36.6

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 36.6

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

   8. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 36.7

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

   10. Applied rewrites 36.7%

       \[\leadsto \frac{y \cdot t}{a} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y}{a} \cdot t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+120}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+204}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* (/ y a) t)))
   (if (<= t_1 -4e+120)
     t_2
     (if (<= t_1 2e-61) x (if (<= t_1 5e+204) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y / a) * t;
	double tmp;
	if (t_1 <= -4e+120) {
		tmp = t_2;
	} else if (t_1 <= 2e-61) {
		tmp = x;
	} else if (t_1 <= 5e+204) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = (y / a) * t
    if (t_1 <= (-4d+120)) then
        tmp = t_2
    else if (t_1 <= 2d-61) then
        tmp = x
    else if (t_1 <= 5d+204) then
        tmp = x + y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y / a) * t;
	double tmp;
	if (t_1 <= -4e+120) {
		tmp = t_2;
	} else if (t_1 <= 2e-61) {
		tmp = x;
	} else if (t_1 <= 5e+204) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = (y / a) * t
	tmp = 0
	if t_1 <= -4e+120:
		tmp = t_2
	elif t_1 <= 2e-61:
		tmp = x
	elif t_1 <= 5e+204:
		tmp = x + y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(y / a) * t)
	tmp = 0.0
	if (t_1 <= -4e+120)
		tmp = t_2;
	elseif (t_1 <= 2e-61)
		tmp = x;
	elseif (t_1 <= 5e+204)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = (y / a) * t;
	tmp = 0.0;
	if (t_1 <= -4e+120)
		tmp = t_2;
	elseif (t_1 <= 2e-61)
		tmp = x;
	elseif (t_1 <= 5e+204)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+120], t$95$2, If[LessEqual[t$95$1, 2e-61], x, If[LessEqual[t$95$1, 5e+204], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.9999999999999999e120 or 5.00000000000000008e204 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 50.1

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

   8. Applied rewrites 50.1%

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

   if -3.9999999999999999e120 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-61

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 72.6%

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

   if 2.0000000000000001e-61 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000008e204

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 84.2%

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

Alternative 5: 71.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot \frac{t}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+120}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+204}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y (/ t a))))
   (if (<= t_1 -4e+120)
     t_2
     (if (<= t_1 2e-61) x (if (<= t_1 5e+204) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * (t / a);
	double tmp;
	if (t_1 <= -4e+120) {
		tmp = t_2;
	} else if (t_1 <= 2e-61) {
		tmp = x;
	} else if (t_1 <= 5e+204) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = y * (t / a)
    if (t_1 <= (-4d+120)) then
        tmp = t_2
    else if (t_1 <= 2d-61) then
        tmp = x
    else if (t_1 <= 5d+204) then
        tmp = x + y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * (t / a);
	double tmp;
	if (t_1 <= -4e+120) {
		tmp = t_2;
	} else if (t_1 <= 2e-61) {
		tmp = x;
	} else if (t_1 <= 5e+204) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = y * (t / a)
	tmp = 0
	if t_1 <= -4e+120:
		tmp = t_2
	elif t_1 <= 2e-61:
		tmp = x
	elif t_1 <= 5e+204:
		tmp = x + y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(y * Float64(t / a))
	tmp = 0.0
	if (t_1 <= -4e+120)
		tmp = t_2;
	elseif (t_1 <= 2e-61)
		tmp = x;
	elseif (t_1 <= 5e+204)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = y * (t / a);
	tmp = 0.0;
	if (t_1 <= -4e+120)
		tmp = t_2;
	elseif (t_1 <= 2e-61)
		tmp = x;
	elseif (t_1 <= 5e+204)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+120], t$95$2, If[LessEqual[t$95$1, 2e-61], x, If[LessEqual[t$95$1, 5e+204], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.9999999999999999e120 or 5.00000000000000008e204 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 50.1

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

   8. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.0

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

   10. Applied rewrites 48.0%

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

   if -3.9999999999999999e120 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-61

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 72.6%

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

   if 2.0000000000000001e-61 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000008e204

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 84.2%

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

Alternative 6: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-163} \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 5e-163) (not (<= t_1 2.0)))
     (fma (/ (- t) (- z a)) y x)
     (fma y (/ z (- z a)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 5e-163) || !(t_1 <= 2.0)) {
		tmp = fma((-t / (z - a)), y, x);
	} else {
		tmp = fma(y, (z / (z - a)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 5e-163) || !(t_1 <= 2.0))
		tmp = fma(Float64(Float64(-t) / Float64(z - a)), y, x);
	else
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e-163], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[((-t) / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999977e-163 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.6

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

   7. Applied rewrites 90.6%

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

   if 4.99999999999999977e-163 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-163} \lor \neg \left(\frac{z - t}{z - a} \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 5e-163)
     (fma t (/ y a) x)
     (if (<= t_1 2e+140) (fma y (/ z (- z a)) x) (* (- t) (/ y (- z a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 5e-163) {
		tmp = fma(t, (y / a), x);
	} else if (t_1 <= 2e+140) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = -t * (y / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 5e-163)
		tmp = fma(t, Float64(y / a), x);
	elseif (t_1 <= 2e+140)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-163], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+140], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999977e-163

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 4.99999999999999977e-163 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000012e140

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if 2.00000000000000012e140 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 83.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 83.0

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.0

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{z - a}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 85.6

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

   10. Applied rewrites 85.6%

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

       1. lift--.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-neg.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift--.f64 85.6

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

   12. Applied rewrites 85.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{\left(-z\right) + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 5e-163)
     (fma t (/ y a) x)
     (if (<= t_1 5e+123) (fma y (/ z (- z a)) x) (/ (* t y) (+ (- z) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 5e-163) {
		tmp = fma(t, (y / a), x);
	} else if (t_1 <= 5e+123) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = (t * y) / (-z + a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 5e-163)
		tmp = fma(t, Float64(y / a), x);
	elseif (t_1 <= 5e+123)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = Float64(Float64(t * y) / Float64(Float64(-z) + a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-163], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+123], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(t * y), $MachinePrecision] / N[((-z) + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999977e-163

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 4.99999999999999977e-163 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999974e123

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if 4.99999999999999974e123 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 84.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{z - a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift--.f64 83.3

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

   5. Applied rewrites 83.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{\left(-z\right) + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 5e-163)
     (fma t (/ y a) x)
     (if (<= t_1 4e+87) (fma y (/ z (- z a)) x) (fma y (/ (- t) z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 5e-163) {
		tmp = fma(t, (y / a), x);
	} else if (t_1 <= 4e+87) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = fma(y, (-t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 5e-163)
		tmp = fma(t, Float64(y / a), x);
	elseif (t_1 <= 4e+87)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = fma(y, Float64(Float64(-t) / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-163], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+87], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999977e-163

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 4.99999999999999977e-163 < (/.f64 (-.f64 z t) (-.f64 z a)) < 3.9999999999999998e87

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 91.8

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

   5. Applied rewrites 91.8%

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

   if 3.9999999999999998e87 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 86.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.0

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

   8. Applied rewrites 70.0%

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

Alternative 10: 79.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+87}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-41)
     (fma (/ t a) y x)
     (if (<= t_1 4e+87) (+ x y) (fma y (/ (- t) z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-41) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 4e+87) {
		tmp = x + y;
	} else {
		tmp = fma(y, (-t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-41)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 4e+87)
		tmp = Float64(x + y);
	else
		tmp = fma(y, Float64(Float64(-t) / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-41], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+87], N[(x + y), $MachinePrecision], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000001e-41

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 81.8

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

   5. Applied rewrites 81.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.8

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

   7. Applied rewrites 81.8%

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

   if 1.00000000000000001e-41 < (/.f64 (-.f64 z t) (-.f64 z a)) < 3.9999999999999998e87

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 89.8%

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

   if 3.9999999999999998e87 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 86.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.0

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

   8. Applied rewrites 70.0%

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

Alternative 11: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+140}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-41)
     (fma (/ t a) y x)
     (if (<= t_1 2e+140) (+ x y) (/ (* y (- t)) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-41) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2e+140) {
		tmp = x + y;
	} else {
		tmp = (y * -t) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-41)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2e+140)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(y * Float64(-t)) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-41], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+140], N[(x + y), $MachinePrecision], N[(N[(y * (-t)), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000001e-41

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 81.8

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

   5. Applied rewrites 81.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.8

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

   7. Applied rewrites 81.8%

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

   if 1.00000000000000001e-41 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000012e140

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 88.8%

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

   if 2.00000000000000012e140 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 83.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.5

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

   8. Applied rewrites 67.5%

      \[\leadsto -\frac{y \cdot t}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+140}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-49} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+123}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 5e-49) (not (<= t_1 5e+123))) (fma t (/ y a) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 5e-49) || !(t_1 <= 5e+123)) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 5e-49) || !(t_1 <= 5e+123))
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e-49], N[Not[LessEqual[t$95$1, 5e+123]], $MachinePrecision]], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 4.9999999999999999e-49 or 4.99999999999999974e123 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if 4.9999999999999999e-49 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999974e123

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 88.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-49} \lor \neg \left(\frac{z - t}{z - a} \leq 5 \cdot 10^{+123}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+123}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-41)
     (fma (/ t a) y x)
     (if (<= t_1 5e+123) (+ x y) (fma t (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-41) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 5e+123) {
		tmp = x + y;
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-41)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 5e+123)
		tmp = Float64(x + y);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-41], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+123], N[(x + y), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000001e-41

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 81.8

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

   5. Applied rewrites 81.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.8

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

   7. Applied rewrites 81.8%

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

   if 1.00000000000000001e-41 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999974e123

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 89.5%

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

   if 4.99999999999999974e123 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 84.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 45.2

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

   5. Applied rewrites 45.2%

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

Alternative 14: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;a \leq 14600:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.1e+37)
   (fma (/ t a) y x)
   (if (<= a 14600.0) (fma y (/ (- z t) z) x) (fma y (/ z (- z a)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.1e+37) {
		tmp = fma((t / a), y, x);
	} else if (a <= 14600.0) {
		tmp = fma(y, ((z - t) / z), x);
	} else {
		tmp = fma(y, (z / (z - a)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3.1e+37)
		tmp = fma(Float64(t / a), y, x);
	elseif (a <= 14600.0)
		tmp = fma(y, Float64(Float64(z - t) / z), x);
	else
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3.1e+37], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[a, 14600.0], N[(y * N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.1000000000000002e37

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 87.3

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

   5. Applied rewrites 87.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 87.3

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

   7. Applied rewrites 87.3%

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

   if -3.1000000000000002e37 < a < 14600

   1. Initial program 96.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if 14600 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 85.8

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

   5. Applied rewrites 85.8%

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

Alternative 15: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 7.8e-53) x (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 7.8e-53) {
		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 (((z - t) / (z - a)) <= 7.8d-53) 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 (((z - t) / (z - a)) <= 7.8e-53) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 7.8e-53:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 7.8e-53)
		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 (((z - t) / (z - a)) <= 7.8e-53)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 7.8e-53], x, N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 7.8000000000000004e-53

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 62.3%

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

   if 7.8000000000000004e-53 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 72.9%

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

Alternative 16: 49.9% 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 97.8%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 56.2%

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

Developer Target 1: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{\frac{z - a}{z - t}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x + (y / ((z - a) / (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, 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 - a) / (z - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025064 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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