Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 84.9% → 97.0%

Time: 9.6s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * (y - z)) / (t - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * (y - z)) / (t - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((y - z) / (t - z)) * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((y - z) / (t - z)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 97.2

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

4. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{y - z}{t - z} \cdot x} \]
5. Add Preprocessing

Alternative 2: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+203}:\\ \;\;\;\;x - \frac{x}{z} \cdot \left(y - t\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.5e+203)
   (- x (* (/ x z) (- y t)))
   (if (<= z 1.15e+90) (* (/ x (- t z)) (- y z)) (* (/ (- z y) z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.5e+203) {
		tmp = x - ((x / z) * (y - t));
	} else if (z <= 1.15e+90) {
		tmp = (x / (t - z)) * (y - z);
	} else {
		tmp = ((z - y) / z) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2.5d+203)) then
        tmp = x - ((x / z) * (y - t))
    else if (z <= 1.15d+90) then
        tmp = (x / (t - z)) * (y - z)
    else
        tmp = ((z - y) / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.5e+203) {
		tmp = x - ((x / z) * (y - t));
	} else if (z <= 1.15e+90) {
		tmp = (x / (t - z)) * (y - z);
	} else {
		tmp = ((z - y) / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.5e+203:
		tmp = x - ((x / z) * (y - t))
	elif z <= 1.15e+90:
		tmp = (x / (t - z)) * (y - z)
	else:
		tmp = ((z - y) / z) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.5e+203)
		tmp = Float64(x - Float64(Float64(x / z) * Float64(y - t)));
	elseif (z <= 1.15e+90)
		tmp = Float64(Float64(x / Float64(t - z)) * Float64(y - z));
	else
		tmp = Float64(Float64(Float64(z - y) / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.5e+203)
		tmp = x - ((x / z) * (y - t));
	elseif (z <= 1.15e+90)
		tmp = (x / (t - z)) * (y - z);
	else
		tmp = ((z - y) / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.5e+203], N[(x - N[(N[(x / z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e+90], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.49999999999999997e203

   1. Initial program 84.0%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if -2.49999999999999997e203 < z < 1.15e90

   1. Initial program 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

   if 1.15e90 < z

   1. Initial program 77.0%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

          \[\leadsto \frac{z + \color{blue}{-1 \cdot y}}{z} \cdot x \]

      16. metadata-eval N/A

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

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

          \[\leadsto \frac{\color{blue}{z - 1 \cdot y}}{z} \cdot x \]

      18. *-lft-identity N/A

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

      19. lower--.f64 92.8

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

   5. Applied rewrites 92.8%

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

Alternative 3: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-21} \lor \neg \left(z \leq 6.7 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7e-21) (not (<= z 6.7e-9)))
   (* (/ (- z y) z) x)
   (* (/ x (- t z)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7e-21) || !(z <= 6.7e-9)) {
		tmp = ((z - y) / z) * x;
	} else {
		tmp = (x / (t - z)) * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-7d-21)) .or. (.not. (z <= 6.7d-9))) then
        tmp = ((z - y) / z) * x
    else
        tmp = (x / (t - z)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7e-21) || !(z <= 6.7e-9)) {
		tmp = ((z - y) / z) * x;
	} else {
		tmp = (x / (t - z)) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7e-21) or not (z <= 6.7e-9):
		tmp = ((z - y) / z) * x
	else:
		tmp = (x / (t - z)) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7e-21) || !(z <= 6.7e-9))
		tmp = Float64(Float64(Float64(z - y) / z) * x);
	else
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -7e-21) || ~((z <= 6.7e-9)))
		tmp = ((z - y) / z) * x;
	else
		tmp = (x / (t - z)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -7e-21], N[Not[LessEqual[z, 6.7e-9]], $MachinePrecision]], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.0000000000000007e-21 or 6.69999999999999961e-9 < z

   1. Initial program 80.7%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

          \[\leadsto \frac{z + \color{blue}{-1 \cdot y}}{z} \cdot x \]

      16. metadata-eval N/A

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

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

          \[\leadsto \frac{\color{blue}{z - 1 \cdot y}}{z} \cdot x \]

      18. *-lft-identity N/A

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

      19. lower--.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -7.0000000000000007e-21 < z < 6.69999999999999961e-9

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-21} \lor \neg \left(z \leq 6.7 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-21}:\\ \;\;\;\;x - \frac{x}{z} \cdot \left(y - t\right)\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7e-21)
   (- x (* (/ x z) (- y t)))
   (if (<= z 6.7e-9) (* (/ x (- t z)) y) (* (/ (- z y) z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7e-21) {
		tmp = x - ((x / z) * (y - t));
	} else if (z <= 6.7e-9) {
		tmp = (x / (t - z)) * y;
	} else {
		tmp = ((z - y) / z) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-7d-21)) then
        tmp = x - ((x / z) * (y - t))
    else if (z <= 6.7d-9) then
        tmp = (x / (t - z)) * y
    else
        tmp = ((z - y) / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7e-21) {
		tmp = x - ((x / z) * (y - t));
	} else if (z <= 6.7e-9) {
		tmp = (x / (t - z)) * y;
	} else {
		tmp = ((z - y) / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7e-21:
		tmp = x - ((x / z) * (y - t))
	elif z <= 6.7e-9:
		tmp = (x / (t - z)) * y
	else:
		tmp = ((z - y) / z) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7e-21)
		tmp = Float64(x - Float64(Float64(x / z) * Float64(y - t)));
	elseif (z <= 6.7e-9)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	else
		tmp = Float64(Float64(Float64(z - y) / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7e-21)
		tmp = x - ((x / z) * (y - t));
	elseif (z <= 6.7e-9)
		tmp = (x / (t - z)) * y;
	else
		tmp = ((z - y) / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7e-21], N[(x - N[(N[(x / z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.7e-9], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.0000000000000007e-21

   1. Initial program 77.1%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if -7.0000000000000007e-21 < z < 6.69999999999999961e-9

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 6.69999999999999961e-9 < z

   1. Initial program 84.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

          \[\leadsto \frac{z + \color{blue}{-1 \cdot y}}{z} \cdot x \]

      16. metadata-eval N/A

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

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

          \[\leadsto \frac{\color{blue}{z - 1 \cdot y}}{z} \cdot x \]

      18. *-lft-identity N/A

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

      19. lower--.f64 80.4

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

   5. Applied rewrites 80.4%

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

Alternative 5: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-21}:\\ \;\;\;\;x - \frac{x}{z} \cdot y\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7e-21)
   (- x (* (/ x z) y))
   (if (<= z 6.7e-9) (* (/ x (- t z)) y) (* (/ (- z y) z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7e-21) {
		tmp = x - ((x / z) * y);
	} else if (z <= 6.7e-9) {
		tmp = (x / (t - z)) * y;
	} else {
		tmp = ((z - y) / z) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-7d-21)) then
        tmp = x - ((x / z) * y)
    else if (z <= 6.7d-9) then
        tmp = (x / (t - z)) * y
    else
        tmp = ((z - y) / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7e-21) {
		tmp = x - ((x / z) * y);
	} else if (z <= 6.7e-9) {
		tmp = (x / (t - z)) * y;
	} else {
		tmp = ((z - y) / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7e-21:
		tmp = x - ((x / z) * y)
	elif z <= 6.7e-9:
		tmp = (x / (t - z)) * y
	else:
		tmp = ((z - y) / z) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7e-21)
		tmp = Float64(x - Float64(Float64(x / z) * y));
	elseif (z <= 6.7e-9)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	else
		tmp = Float64(Float64(Float64(z - y) / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7e-21)
		tmp = x - ((x / z) * y);
	elseif (z <= 6.7e-9)
		tmp = (x / (t - z)) * y;
	else
		tmp = ((z - y) / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7e-21], N[(x - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.7e-9], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.0000000000000007e-21

   1. Initial program 77.1%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 82.3%

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

   if -7.0000000000000007e-21 < z < 6.69999999999999961e-9

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 6.69999999999999961e-9 < z

   1. Initial program 84.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. mul-1-neg N/A

          \[\leadsto \frac{z + \color{blue}{-1 \cdot y}}{z} \cdot x \]

      16. metadata-eval N/A

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

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

          \[\leadsto \frac{\color{blue}{z - 1 \cdot y}}{z} \cdot x \]

      18. *-lft-identity N/A

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

      19. lower--.f64 80.4

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

   5. Applied rewrites 80.4%

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

Alternative 6: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.1e+32)
   (fma (/ x z) t x)
   (if (<= z 2.3e+114) (* (/ x (- t z)) y) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.1e+32) {
		tmp = fma((x / z), t, x);
	} else if (z <= 2.3e+114) {
		tmp = (x / (t - z)) * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.1e+32)
		tmp = fma(Float64(x / z), t, x);
	elseif (z <= 2.3e+114)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.1e+32], N[(N[(x / z), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 2.3e+114], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.09999999999999993e32

   1. Initial program 73.8%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.0%

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

   if -3.09999999999999993e32 < z < 2.3e114

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if 2.3e114 < z

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 86.4%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{t} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.2e-21)
   (fma (/ x z) t x)
   (if (<= z 2.05e-6) (* (/ x t) (- y z)) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.2e-21) {
		tmp = fma((x / z), t, x);
	} else if (z <= 2.05e-6) {
		tmp = (x / t) * (y - z);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.2e-21)
		tmp = fma(Float64(x / z), t, x);
	elseif (z <= 2.05e-6)
		tmp = Float64(Float64(x / t) * Float64(y - z));
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.2e-21], N[(N[(x / z), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 2.05e-6], N[(N[(x / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.19999999999999988e-21

   1. Initial program 77.1%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.1%

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

   if -8.19999999999999988e-21 < z < 2.0499999999999999e-6

   1. Initial program 94.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   7. Applied rewrites 79.1%

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

   if 2.0499999999999999e-6 < z

   1. Initial program 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 60.8%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-19} \lor \neg \left(z \leq 85000000000\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.04e-19) (not (<= z 85000000000.0))) (* 1.0 x) (* (/ y t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.04e-19) || !(z <= 85000000000.0)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y / t) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.04d-19)) .or. (.not. (z <= 85000000000.0d0))) then
        tmp = 1.0d0 * x
    else
        tmp = (y / t) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.04e-19) || !(z <= 85000000000.0)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y / t) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.04e-19) or not (z <= 85000000000.0):
		tmp = 1.0 * x
	else:
		tmp = (y / t) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.04e-19) || !(z <= 85000000000.0))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(y / t) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.04e-19) || ~((z <= 85000000000.0)))
		tmp = 1.0 * x;
	else
		tmp = (y / t) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.04e-19], N[Not[LessEqual[z, 85000000000.0]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.03999999999999998e-19 or 8.5e10 < z

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 62.3%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -1.03999999999999998e-19 < z < 8.5e10

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.9%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-19} \lor \neg \left(z \leq 85000000000\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-19} \lor \neg \left(z \leq 6.7 \cdot 10^{-9}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.04e-19) (not (<= z 6.7e-9))) (* 1.0 x) (* (/ x t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.04e-19) || !(z <= 6.7e-9)) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / t) * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.04d-19)) .or. (.not. (z <= 6.7d-9))) then
        tmp = 1.0d0 * x
    else
        tmp = (x / t) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.04e-19) || !(z <= 6.7e-9)) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / t) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.04e-19) or not (z <= 6.7e-9):
		tmp = 1.0 * x
	else:
		tmp = (x / t) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.04e-19) || !(z <= 6.7e-9))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(x / t) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.04e-19) || ~((z <= 6.7e-9)))
		tmp = 1.0 * x;
	else
		tmp = (x / t) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.04e-19], N[Not[LessEqual[z, 6.7e-9]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(x / t), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.03999999999999998e-19 or 6.69999999999999961e-9 < z

   1. Initial program 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 60.7%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -1.03999999999999998e-19 < z < 6.69999999999999961e-9

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 70.3%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-19} \lor \neg \left(z \leq 6.7 \cdot 10^{-9}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 85000000000:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.04e-19)
   (fma (/ x z) t x)
   (if (<= z 85000000000.0) (* (/ y t) x) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.04e-19) {
		tmp = fma((x / z), t, x);
	} else if (z <= 85000000000.0) {
		tmp = (y / t) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.04e-19)
		tmp = fma(Float64(x / z), t, x);
	elseif (z <= 85000000000.0)
		tmp = Float64(Float64(y / t) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.04e-19], N[(N[(x / z), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 85000000000.0], N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.03999999999999998e-19

   1. Initial program 77.1%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.1%

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

   if -1.03999999999999998e-19 < z < 8.5e10

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.9%

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

   if 8.5e10 < z

   1. Initial program 83.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 64.4%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 61.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, x, x\right)\\ \mathbf{elif}\;z \leq 85000000000:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.04e-19)
   (fma (/ t z) x x)
   (if (<= z 85000000000.0) (* (/ y t) x) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.04e-19) {
		tmp = fma((t / z), x, x);
	} else if (z <= 85000000000.0) {
		tmp = (y / t) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.04e-19)
		tmp = fma(Float64(t / z), x, x);
	elseif (z <= 85000000000.0)
		tmp = Float64(Float64(y / t) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.04e-19], N[(N[(t / z), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[z, 85000000000.0], N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.03999999999999998e-19

   1. Initial program 77.1%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 3.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 62.1%

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

   if -1.03999999999999998e-19 < z < 8.5e10

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.9%

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

   if 8.5e10 < z

   1. Initial program 83.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 64.4%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 34.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t) {
	return 1.0 * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Python

def code(x, y, z, t):
	return 1.0 * x

Julia

function code(x, y, z, t)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 97.2

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in z around inf

   \[\leadsto \color{blue}{1} \cdot x \]
6. Step-by-step derivation

7. Applied rewrites 34.0%

   \[\leadsto \color{blue}{1} \cdot x \]
8. Add Preprocessing

Developer Target 1: 97.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / ((t - z) / (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024352 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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