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

Percentage Accurate: 97.0% → 97.0%

Time: 2.9s

Alternatives: 13

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, 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 97.0%

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

Alternative 2: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -5000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{-y}{z - y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -5000000000000.0)
     t_2
     (if (<= t_1 0.0001)
       (* (/ (- x y) z) t)
       (if (<= t_1 2.0) (* (/ (- y) (- z 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 <= -5000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0001) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = (-y / (z - y)) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = (x / (z - y)) * t
    if (t_1 <= (-5000000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.0001d0) then
        tmp = ((x - y) / z) * t
    else if (t_1 <= 2.0d0) then
        tmp = (-y / (z - y)) * t
    else
        tmp = t_2
    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 t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0001) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = (-y / (z - y)) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = (x / (z - y)) * t
	tmp = 0
	if t_1 <= -5000000000000.0:
		tmp = t_2
	elif t_1 <= 0.0001:
		tmp = ((x - y) / z) * t
	elif t_1 <= 2.0:
		tmp = (-y / (z - 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 <= -5000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(Float64(-y) / Float64(z - y)) * t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = (x / (z - y)) * t;
	tmp = 0.0;
	if (t_1 <= -5000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0001)
		tmp = ((x - y) / z) * t;
	elseif (t_1 <= 2.0)
		tmp = (-y / (z - y)) * t;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000000.0], t$95$2, If[LessEqual[t$95$1, 0.0001], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[((-y) / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e12 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 94.6%

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

   if -5e12 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000005e-4

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 92.9%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.0

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

   4. Applied rewrites 98.0%

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

Alternative 3: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -5000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -5000000000000.0)
     t_2
     (if (<= t_1 0.0001)
       (* (/ (- x y) z) t)
       (if (<= t_1 2.0) (fma t (/ z 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 <= -5000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0001) {
		tmp = ((x - y) / z) * t;
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / 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 <= -5000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(Float64(x - y) / z) * t);
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / 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, -5000000000000.0], t$95$2, If[LessEqual[t$95$1, 0.0001], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e12 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 94.6%

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

   if -5e12 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000005e-4

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 92.9%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto t + \color{blue}{\frac{t \cdot z}{y}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lift-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 4: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-11}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -1e-11)
     t_2
     (if (<= t_1 0.0001)
       (* (- x y) (/ t z))
       (if (<= t_1 2.0) (fma t (/ z 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 <= -1e-11) {
		tmp = t_2;
	} else if (t_1 <= 0.0001) {
		tmp = (x - y) * (t / z);
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / 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 <= -1e-11)
		tmp = t_2;
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(x - y) * Float64(t / z));
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / 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, -1e-11], t$95$2, If[LessEqual[t$95$1, 0.0001], N[(N[(x - y), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -9.99999999999999939e-12 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 92.5%

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

   if -9.99999999999999939e-12 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000005e-4

   1. Initial program 95.6%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 72.4

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 88.3%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lower-/.f64 90.2

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

   11. Applied rewrites 90.2%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto t + \color{blue}{\frac{t \cdot z}{y}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lift-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 5: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := x \cdot \frac{t}{z - y}\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* x (/ t (- z y)))))
   (if (<= t_1 -50000000.0)
     t_2
     (if (<= t_1 0.0001)
       (* (- x y) (/ t z))
       (if (<= t_1 2.0) (fma t (/ z 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 * (t / (z - y));
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0001) {
		tmp = (x - y) * (t / z);
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / 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(x * Float64(t / Float64(z - y)))
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(x - y) * Float64(t / z));
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / 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[(x * N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, 0.0001], N[(N[(x - y), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e7 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 88.7

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

   6. Applied rewrites 88.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 89.9

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

   8. Applied rewrites 89.9%

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

   if -5e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000005e-4

   1. Initial program 95.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 72.3

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

   6. Applied rewrites 72.3%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 86.4%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lower-/.f64 88.2

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

   11. Applied rewrites 88.2%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto t + \color{blue}{\frac{t \cdot z}{y}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lift-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 6: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := x \cdot \frac{t}{z - y}\\ \mathbf{if}\;t\_1 \leq -5000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* x (/ t (- z y)))))
   (if (<= t_1 -5000000000000.0)
     t_2
     (if (<= t_1 0.0001)
       (/ (* (- x y) t) z)
       (if (<= t_1 2.0) (fma t (/ z 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 * (t / (z - y));
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0001) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / 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(x * Float64(t / Float64(z - y)))
	tmp = 0.0
	if (t_1 <= -5000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / 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[(x * N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000000.0], t$95$2, If[LessEqual[t$95$1, 0.0001], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e12 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 89.0

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

   6. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 90.1

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

   8. Applied rewrites 90.1%

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

   if -5e12 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000005e-4

   1. Initial program 95.8%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(x - y\right)}{z}} \]
   3. 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. lift--.f64 85.9

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

   4. Applied rewrites 85.9%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto t + \color{blue}{\frac{t \cdot z}{y}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lift-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 7: 79.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\left(-t\right) \cdot x}{y}\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -5e+130)
     (/ (* (- t) x) y)
     (if (<= t_1 0.0001)
       (/ (* (- x y) t) z)
       (if (<= t_1 2.0) (fma t (/ z y) 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+130) {
		tmp = (-t * x) / y;
	} else if (t_1 <= 0.0001) {
		tmp = ((x - y) * t) / z;
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / y), 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+130)
		tmp = Float64(Float64(Float64(-t) * x) / y);
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(Float64(x - y) * t) / z);
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), 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+130], N[(N[((-t) * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999996e130

   1. Initial program 90.4%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 59.2

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

   7. Applied rewrites 59.2%

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

   if -4.9999999999999996e130 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000005e-4

   1. Initial program 96.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(x - y\right)}{z}} \]
   3. 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. lift--.f64 79.1

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

   4. Applied rewrites 79.1%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto t + \color{blue}{\frac{t \cdot z}{y}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lift-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

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

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 56.3

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

   4. Applied rewrites 56.3%

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

Alternative 8: 70.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ t_2 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\left(-t\right) \cdot x}{y}\\ \mathbf{elif}\;t\_2 \leq 0.0001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)) (t_2 (/ (- x y) (- z y))))
   (if (<= t_2 -5e+130)
     (/ (* (- t) x) y)
     (if (<= t_2 0.0001) t_1 (if (<= t_2 2.0) (fma t (/ z y) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5e+130) {
		tmp = (-t * x) / y;
	} else if (t_2 <= 0.0001) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5e+130)
		tmp = Float64(Float64(Float64(-t) * x) / y);
	elseif (t_2 <= 0.0001)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+130], N[(N[((-t) * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, 0.0001], t$95$1, If[LessEqual[t$95$2, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999996e130

   1. Initial program 90.4%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 59.2

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

   7. Applied rewrites 59.2%

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

   if -4.9999999999999996e130 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000005e-4 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 58.2

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

   4. Applied rewrites 58.2%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto t + \color{blue}{\frac{t \cdot z}{y}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lift-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 9: 70.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x z) t)))
   (if (<= t_1 0.0001) t_2 (if (<= t_1 2.0) (fma t (/ z 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) * t;
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / 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 / z) * t)
	tmp = 0.0
	if (t_1 <= 0.0001)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / 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 / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0001], t$95$2, If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.6%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 58.0

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

   4. Applied rewrites 58.0%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   3. 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. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto t + \color{blue}{\frac{t \cdot z}{y}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lift-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 10: 70.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z} \cdot t\\ \mathbf{if}\;t\_1 \leq 0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x z) t)))
   (if (<= t_1 0.0001) t_2 (if (<= t_1 2.0) 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) * t;
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = (x / z) * t
    if (t_1 <= 0.0001d0) then
        tmp = t_2
    else if (t_1 <= 2.0d0) then
        tmp = t
    else
        tmp = t_2
    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 t_2 = (x / z) * t;
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = (x / z) * t
	tmp = 0
	if t_1 <= 0.0001:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = 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 / z) * t)
	tmp = 0.0
	if (t_1 <= 0.0001)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = (x / z) * t;
	tmp = 0.0;
	if (t_1 <= 0.0001)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0001], t$95$2, If[LessEqual[t$95$1, 2.0], t, t$95$2]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.6%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 58.0

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

   4. Applied rewrites 58.0%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 95.8%

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

Alternative 11: 68.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t\_1 \leq 0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* x (/ t z))))
   (if (<= t_1 0.0001) t_2 (if (<= t_1 2.0) 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 * (t / z);
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = x * (t / z)
    if (t_1 <= 0.0001d0) then
        tmp = t_2
    else if (t_1 <= 2.0d0) then
        tmp = t
    else
        tmp = t_2
    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 t_2 = x * (t / z);
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = x * (t / z)
	tmp = 0
	if t_1 <= 0.0001:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = 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(x * Float64(t / z))
	tmp = 0.0
	if (t_1 <= 0.0001)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = x * (t / z);
	tmp = 0.0;
	if (t_1 <= 0.0001)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0001], t$95$2, If[LessEqual[t$95$1, 2.0], t, t$95$2]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 72.3

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

   6. Applied rewrites 72.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 73.2

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 54.9%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 95.8%

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

Alternative 12: 67.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{t \cdot x}{z}\\ \mathbf{if}\;t\_1 \leq 0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+15}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (/ (* t x) z)))
   (if (<= t_1 0.0001) t_2 (if (<= t_1 1e+15) 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 * x) / z;
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = t_2;
	} else if (t_1 <= 1e+15) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = (t * x) / z
    if (t_1 <= 0.0001d0) then
        tmp = t_2
    else if (t_1 <= 1d+15) then
        tmp = t
    else
        tmp = t_2
    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 t_2 = (t * x) / z;
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = t_2;
	} else if (t_1 <= 1e+15) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = (t * x) / z
	tmp = 0
	if t_1 <= 0.0001:
		tmp = t_2
	elif t_1 <= 1e+15:
		tmp = 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 * x) / z)
	tmp = 0.0
	if (t_1 <= 0.0001)
		tmp = t_2;
	elseif (t_1 <= 1e+15)
		tmp = t;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = (t * x) / z;
	tmp = 0.0;
	if (t_1 <= 0.0001)
		tmp = t_2;
	elseif (t_1 <= 1e+15)
		tmp = t;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0001], t$95$2, If[LessEqual[t$95$1, 1e+15], t, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000005e-4 or 1e15 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 95.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.6

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

   4. Applied rewrites 54.6%

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

   if 1.00000000000000005e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1e15

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 92.4%

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

Alternative 13: 34.1% 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 97.0%

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

2. Taylor expanded in y around inf

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

4. Applied rewrites 34.1%

   \[\leadsto \color{blue}{t} \]
5. 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 2025106 
(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))