Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Percentage Accurate: 97.0% → 97.0%

Time: 6.2s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

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

Alternative 2: 94.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{x \cdot t}{z - y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;t - \frac{x - z}{y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -1e+17)
     (/ (* x t) (- z y))
     (if (<= t_1 5e-8)
       (* (/ (- x y) z) t)
       (if (<= t_1 50000.0) (- t (* (/ (- x z) y) t)) (* (/ x (- z y)) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+17) {
		tmp = (x * t) / (z - y);
	} else if (t_1 <= 5e-8) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 50000.0) {
		tmp = t - (((x - z) / y) * t);
	} else {
		tmp = (x / (z - y)) * t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if (t_1 <= (-1d+17)) then
        tmp = (x * t) / (z - y)
    else if (t_1 <= 5d-8) then
        tmp = ((x - y) / z) * t
    else if (t_1 <= 50000.0d0) then
        tmp = t - (((x - z) / y) * t)
    else
        tmp = (x / (z - y)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+17) {
		tmp = (x * t) / (z - y);
	} else if (t_1 <= 5e-8) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 50000.0) {
		tmp = t - (((x - z) / y) * t);
	} else {
		tmp = (x / (z - y)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if t_1 <= -1e+17:
		tmp = (x * t) / (z - y)
	elif t_1 <= 5e-8:
		tmp = ((x - y) / z) * t
	elif t_1 <= 50000.0:
		tmp = t - (((x - z) / y) * t)
	else:
		tmp = (x / (z - y)) * t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -1e+17)
		tmp = Float64(Float64(x * t) / Float64(z - y));
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 50000.0)
		tmp = Float64(t - Float64(Float64(Float64(x - z) / y) * t));
	else
		tmp = Float64(Float64(x / Float64(z - y)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_1 <= -1e+17)
		tmp = (x * t) / (z - y);
	elseif (t_1 <= 5e-8)
		tmp = ((x - y) / z) * t;
	elseif (t_1 <= 50000.0)
		tmp = t - (((x - z) / y) * t);
	else
		tmp = (x / (z - y)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+17], N[(N[(x * t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-8], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 50000.0], N[(t - N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e17

   1. Initial program 90.1%

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 92.4

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

   9. Applied rewrites 92.4%

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

   if -1e17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 96.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 93.7%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. lower--.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.5%

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

Alternative 3: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{x \cdot t}{z - y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -1e+17)
     (/ (* x t) (- z y))
     (if (<= t_1 5e-8)
       (* (/ (- x y) z) t)
       (if (<= t_1 50000.0) (fma (- t) (/ x y) t) (* (/ x (- z y)) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+17) {
		tmp = (x * t) / (z - y);
	} else if (t_1 <= 5e-8) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 50000.0) {
		tmp = fma(-t, (x / y), t);
	} else {
		tmp = (x / (z - y)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -1e+17)
		tmp = Float64(Float64(x * t) / Float64(z - y));
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 50000.0)
		tmp = fma(Float64(-t), Float64(x / y), t);
	else
		tmp = Float64(Float64(x / Float64(z - y)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+17], N[(N[(x * t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-8], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 50000.0], N[((-t) * N[(x / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e17

   1. Initial program 90.1%

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 92.4

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

   9. Applied rewrites 92.4%

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

   if -1e17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 96.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 93.7%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.5%

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

Alternative 4: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -100:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -100.0)
     t_2
     (if (<= t_1 5e-8)
       (* (/ (- x y) z) t)
       (if (<= t_1 50000.0) (fma (- t) (/ x y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -100.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-8) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 50000.0) {
		tmp = fma(-t, (x / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -100.0)
		tmp = t_2;
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 50000.0)
		tmp = fma(Float64(-t), Float64(x / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -100.0], t$95$2, If[LessEqual[t$95$1, 5e-8], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 50000.0], N[((-t) * N[(x / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -100 or 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.4%

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

   if -100 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 96.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.5%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 97.7

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

   8. Applied rewrites 97.7%

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

Alternative 5: 93.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -100:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -100.0)
     t_2
     (if (<= t_1 5e-8)
       (* (/ t z) (- x y))
       (if (<= t_1 50000.0) (fma (- t) (/ x y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -100.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-8) {
		tmp = (t / z) * (x - y);
	} else if (t_1 <= 50000.0) {
		tmp = fma(-t, (x / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -100.0)
		tmp = t_2;
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(t / z) * Float64(x - y));
	elseif (t_1 <= 50000.0)
		tmp = fma(Float64(-t), Float64(x / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -100.0], t$95$2, If[LessEqual[t$95$1, 5e-8], N[(N[(t / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000.0], N[((-t) * N[(x / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -100 or 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.4%

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

   if -100 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 96.4%

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 87.0%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 97.7

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

   8. Applied rewrites 97.7%

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

Alternative 6: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y} \cdot x\\ \mathbf{if}\;t\_1 \leq -100:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -100.0)
     t_2
     (if (<= t_1 5e-8)
       (* (/ t z) (- x y))
       (if (<= t_1 2e+39) (fma (- t) (/ x y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -100.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-8) {
		tmp = (t / z) * (x - y);
	} else if (t_1 <= 2e+39) {
		tmp = fma(-t, (x / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -100.0)
		tmp = t_2;
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(t / z) * Float64(x - y));
	elseif (t_1 <= 2e+39)
		tmp = fma(Float64(-t), Float64(x / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -100.0], t$95$2, If[LessEqual[t$95$1, 5e-8], N[(N[(t / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+39], N[((-t) * N[(x / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -100 or 1.99999999999999988e39 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if -100 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 96.4%

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 87.0%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999988e39

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 94.0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 93.4

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

   8. Applied rewrites 93.4%

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

Alternative 7: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y} \cdot x\\ \mathbf{if}\;t\_1 \leq -0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -0.0001)
     t_2
     (if (<= t_1 5e-8)
       (/ (* (- x y) t) z)
       (if (<= t_1 2e+39) (fma (- t) (/ x y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -0.0001) {
		tmp = t_2;
	} else if (t_1 <= 5e-8) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 2e+39) {
		tmp = fma(-t, (x / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -0.0001)
		tmp = t_2;
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 2e+39)
		tmp = fma(Float64(-t), Float64(x / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -0.0001], t$95$2, If[LessEqual[t$95$1, 5e-8], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 2e+39], N[((-t) * N[(x / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.00000000000000005e-4 or 1.99999999999999988e39 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if -1.00000000000000005e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 96.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999988e39

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 94.0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 93.4

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

   8. Applied rewrites 93.4%

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

Alternative 8: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t}{z - y} \cdot x\\ \mathbf{if}\;t\_1 \leq -0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{t}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ t (- z y)) x)))
   (if (<= t_1 -0.0001)
     t_2
     (if (<= t_1 5e-8)
       (/ (* (- x y) t) z)
       (if (<= t_1 50000.0) (fma (- x) (/ t y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (t / (z - y)) * x;
	double tmp;
	if (t_1 <= -0.0001) {
		tmp = t_2;
	} else if (t_1 <= 5e-8) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 50000.0) {
		tmp = fma(-x, (t / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(t / Float64(z - y)) * x)
	tmp = 0.0
	if (t_1 <= -0.0001)
		tmp = t_2;
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 50000.0)
		tmp = fma(Float64(-x), Float64(t / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -0.0001], t$95$2, If[LessEqual[t$95$1, 5e-8], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 50000.0], N[((-x) * N[(t / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.00000000000000005e-4 or 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if -1.00000000000000005e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 96.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto -1 \cdot \frac{t \cdot x}{y} + t \]

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 95.9

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

   8. Applied rewrites 95.9%

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

Alternative 9: 70.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{t}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -1e+32)
     (* (/ (- x) y) t)
     (if (<= t_1 5e-17)
       (* (/ x z) t)
       (if (<= t_1 1e+162) (fma (- x) (/ t y) t) (/ (* t x) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+32) {
		tmp = (-x / y) * t;
	} else if (t_1 <= 5e-17) {
		tmp = (x / z) * t;
	} else if (t_1 <= 1e+162) {
		tmp = fma(-x, (t / y), t);
	} else {
		tmp = (t * x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -1e+32)
		tmp = Float64(Float64(Float64(-x) / y) * t);
	elseif (t_1 <= 5e-17)
		tmp = Float64(Float64(x / z) * t);
	elseif (t_1 <= 1e+162)
		tmp = fma(Float64(-x), Float64(t / y), t);
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+32], N[(N[((-x) / y), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 5e-17], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 1e+162], N[((-x) * N[(t / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.00000000000000005e32

   1. Initial program 88.9%

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

   3. Taylor expanded in z around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-out N/A

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

      3. metadata-eval N/A

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

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

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-lft-neg-out N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. div-sub N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. fp-cancel-sign-sub N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. metadata-eval N/A

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

      20. distribute-lft-neg-out N/A

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

      21. metadata-eval N/A

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

      22. distribute-neg-in N/A

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

      23. +-commutative N/A

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

      24. distribute-neg-in N/A

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

      25. metadata-eval N/A

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

      26. distribute-lft-neg-out N/A

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

      27. metadata-eval N/A

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

      28. *-lft-identity N/A

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{y} \cdot t \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x}{y} \cdot t \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 66.5

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

   8. Applied rewrites 66.5%

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

   if -1.00000000000000005e32 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if 4.9999999999999999e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999994e161

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto -1 \cdot \frac{t \cdot x}{y} + t \]

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 82.9

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

   8. Applied rewrites 82.9%

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

   if 9.9999999999999994e161 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

Alternative 10: 70.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_1 \leq 50000\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -1e+32)
     (* (/ (- x) y) t)
     (if (or (<= t_1 5e-8) (not (<= t_1 50000.0)))
       (* (/ x z) t)
       (fma (/ z y) t t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+32) {
		tmp = (-x / y) * t;
	} else if ((t_1 <= 5e-8) || !(t_1 <= 50000.0)) {
		tmp = (x / z) * t;
	} else {
		tmp = fma((z / y), t, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -1e+32)
		tmp = Float64(Float64(Float64(-x) / y) * t);
	elseif ((t_1 <= 5e-8) || !(t_1 <= 50000.0))
		tmp = Float64(Float64(x / z) * t);
	else
		tmp = fma(Float64(z / y), t, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+32], N[(N[((-x) / y), $MachinePrecision] * t), $MachinePrecision], If[Or[LessEqual[t$95$1, 5e-8], N[Not[LessEqual[t$95$1, 50000.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * t + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.00000000000000005e32

   1. Initial program 88.9%

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

   3. Taylor expanded in z around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-out N/A

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

      3. metadata-eval N/A

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

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

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-lft-neg-out N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. div-sub N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. fp-cancel-sign-sub N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. metadata-eval N/A

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

      20. distribute-lft-neg-out N/A

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

      21. metadata-eval N/A

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

      22. distribute-neg-in N/A

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

      23. +-commutative N/A

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

      24. distribute-neg-in N/A

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

      25. metadata-eval N/A

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

      26. distribute-lft-neg-out N/A

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

      27. metadata-eval N/A

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

      28. *-lft-identity N/A

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{y} \cdot t \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x}{y} \cdot t \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 66.5

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

   8. Applied rewrites 66.5%

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

   if -1.00000000000000005e32 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8 or 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 92.3

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

   8. Applied rewrites 92.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-8} \lor \neg \left(\frac{x - y}{z - y} \leq 50000\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-t}{y} \cdot x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_1 \leq 50000\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -1e+32)
     (* (/ (- t) y) x)
     (if (or (<= t_1 5e-8) (not (<= t_1 50000.0)))
       (* (/ x z) t)
       (fma (/ z y) t t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -1e+32) {
		tmp = (-t / y) * x;
	} else if ((t_1 <= 5e-8) || !(t_1 <= 50000.0)) {
		tmp = (x / z) * t;
	} else {
		tmp = fma((z / y), t, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -1e+32)
		tmp = Float64(Float64(Float64(-t) / y) * x);
	elseif ((t_1 <= 5e-8) || !(t_1 <= 50000.0))
		tmp = Float64(Float64(x / z) * t);
	else
		tmp = fma(Float64(z / y), t, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+32], N[(N[((-t) / y), $MachinePrecision] * x), $MachinePrecision], If[Or[LessEqual[t$95$1, 5e-8], N[Not[LessEqual[t$95$1, 50000.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * t + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.00000000000000005e32

   1. Initial program 88.9%

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 63.5

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

   10. Applied rewrites 63.5%

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

   if -1.00000000000000005e32 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8 or 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 92.3

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

   8. Applied rewrites 92.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-t}{y} \cdot x\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-8} \lor \neg \left(\frac{x - y}{z - y} \leq 50000\right):\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-17} \lor \neg \left(t\_1 \leq 50000\right):\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{t}{y}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (or (<= t_1 5e-17) (not (<= t_1 50000.0)))
     (* (/ t (- z y)) x)
     (fma (- x) (/ t y) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 5e-17) || !(t_1 <= 50000.0)) {
		tmp = (t / (z - y)) * x;
	} else {
		tmp = fma(-x, (t / y), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_1 <= 5e-17) || !(t_1 <= 50000.0))
		tmp = Float64(Float64(t / Float64(z - y)) * x);
	else
		tmp = fma(Float64(-x), Float64(t / y), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e-17], N[Not[LessEqual[t$95$1, 50000.0]], $MachinePrecision]], N[(N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[((-x) * N[(t / y), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17 or 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if 4.9999999999999999e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto -1 \cdot \frac{t \cdot x}{y} + t \]

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 94.9

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

   8. Applied rewrites 94.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-17} \lor \neg \left(\frac{x - y}{z - y} \leq 50000\right):\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{t}{y}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 5e-8)
     (* (/ t z) x)
     (if (<= t_1 50000.0) (fma (/ z y) t t) (* (/ x z) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 5e-8) {
		tmp = (t / z) * x;
	} else if (t_1 <= 50000.0) {
		tmp = fma((z / y), t, t);
	} else {
		tmp = (x / z) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= 5e-8)
		tmp = Float64(Float64(t / z) * x);
	elseif (t_1 <= 50000.0)
		tmp = fma(Float64(z / y), t, t);
	else
		tmp = Float64(Float64(x / z) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-8], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 50000.0], N[(N[(z / y), $MachinePrecision] * t + t), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 94.6%

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 51.0

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

   10. Applied rewrites 51.0%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 92.3

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

   8. Applied rewrites 92.3%

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

   if 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 59.3

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

   5. Applied rewrites 59.3%

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

Alternative 14: 69.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-17} \lor \neg \left(t\_1 \leq 50000\right):\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (or (<= t_1 5e-17) (not (<= t_1 50000.0))) (* (/ t z) x) t)))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 5e-17) || !(t_1 <= 50000.0)) {
		tmp = (t / z) * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if ((t_1 <= 5d-17) .or. (.not. (t_1 <= 50000.0d0))) then
        tmp = (t / z) * x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 5e-17) || !(t_1 <= 50000.0)) {
		tmp = (t / z) * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if (t_1 <= 5e-17) or not (t_1 <= 50000.0):
		tmp = (t / z) * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_1 <= 5e-17) || !(t_1 <= 50000.0))
		tmp = Float64(Float64(t / z) * x);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_1 <= 5e-17) || ~((t_1 <= 50000.0)))
		tmp = (t / z) * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e-17], N[Not[LessEqual[t$95$1, 50000.0]], $MachinePrecision]], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17 or 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.3%

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 71.2%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 52.1

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

   10. Applied rewrites 52.1%

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

   if 4.9999999999999999e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 90.7%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-17} \lor \neg \left(\frac{x - y}{z - y} \leq 50000\right):\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-58} \lor \neg \left(t\_1 \leq 50000\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (or (<= t_1 2e-58) (not (<= t_1 50000.0))) (/ (* t x) z) t)))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 2e-58) || !(t_1 <= 50000.0)) {
		tmp = (t * x) / z;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if ((t_1 <= 2d-58) .or. (.not. (t_1 <= 50000.0d0))) then
        tmp = (t * x) / z
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if ((t_1 <= 2e-58) || !(t_1 <= 50000.0)) {
		tmp = (t * x) / z;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if (t_1 <= 2e-58) or not (t_1 <= 50000.0):
		tmp = (t * x) / z
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_1 <= 2e-58) || !(t_1 <= 50000.0))
		tmp = Float64(Float64(t * x) / z);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_1 <= 2e-58) || ~((t_1 <= 50000.0)))
		tmp = (t * x) / z;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 2e-58], N[Not[LessEqual[t$95$1, 50000.0]], $MachinePrecision]], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-58 or 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

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

   if 2.0000000000000001e-58 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 81.4%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-58} \lor \neg \left(\frac{x - y}{z - y} \leq 50000\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 5e-17) (* (/ t z) x) (if (<= t_1 50000.0) t (* (/ x z) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 5e-17) {
		tmp = (t / z) * x;
	} else if (t_1 <= 50000.0) {
		tmp = t;
	} else {
		tmp = (x / z) * t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    if (t_1 <= 5d-17) then
        tmp = (t / z) * x
    else if (t_1 <= 50000.0d0) then
        tmp = t
    else
        tmp = (x / z) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= 5e-17) {
		tmp = (t / z) * x;
	} else if (t_1 <= 50000.0) {
		tmp = t;
	} else {
		tmp = (x / z) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	tmp = 0
	if t_1 <= 5e-17:
		tmp = (t / z) * x
	elif t_1 <= 50000.0:
		tmp = t
	else:
		tmp = (x / z) * t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= 5e-17)
		tmp = Float64(Float64(t / z) * x);
	elseif (t_1 <= 50000.0)
		tmp = t;
	else
		tmp = Float64(Float64(x / z) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_1 <= 5e-17)
		tmp = (t / z) * x;
	elseif (t_1 <= 50000.0)
		tmp = t;
	else
		tmp = (x / z) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-17], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 50000.0], t, N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999999e-17

   1. Initial program 94.6%

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 65.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 51.3

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

   10. Applied rewrites 51.3%

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

   if 4.9999999999999999e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 90.7%

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

   if 5e4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 59.3

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

   5. Applied rewrites 59.3%

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

Alternative 17: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

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

Alternative 18: 35.1% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 96.7%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 30.8%

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

Developer Target 1: 97.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025026 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

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

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