Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 14

Speedup: 1.0×

• 33.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;x - \left(t - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.5e+15)
   (fma (- t x) y x)
   (if (<= y 1.3e-5) (- x (* (- t x) z)) (* (- t x) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.5e+15) {
		tmp = fma((t - x), y, x);
	} else if (y <= 1.3e-5) {
		tmp = x - ((t - x) * z);
	} else {
		tmp = (t - x) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.5e+15)
		tmp = fma(Float64(t - x), y, x);
	elseif (y <= 1.3e-5)
		tmp = Float64(x - Float64(Float64(t - x) * z));
	else
		tmp = Float64(Float64(t - x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6.5e+15], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 1.3e-5], N[(x - N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5e15

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(t - x\right) \cdot y + x \]

      3. lower-fma.f64 N/A

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

      4. lift--.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -6.5e15 < y < 1.29999999999999992e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 91.5

         \[\leadsto x - \left(t - x\right) \cdot z \]

   5. Applied rewrites 91.5%

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

   if 1.29999999999999992e-5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 85.6

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

   5. Applied rewrites 85.6%

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

Alternative 3: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \left(y - z\right)\right) \cdot x\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-34}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (- y z)) x)))
   (if (<= x -6.5e-59) t_1 (if (<= x 1.4e-34) (* (- y z) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - (y - z)) * x;
	double tmp;
	if (x <= -6.5e-59) {
		tmp = t_1;
	} else if (x <= 1.4e-34) {
		tmp = (y - z) * t;
	} else {
		tmp = t_1;
	}
	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 = (1.0d0 - (y - z)) * x
    if (x <= (-6.5d-59)) then
        tmp = t_1
    else if (x <= 1.4d-34) then
        tmp = (y - z) * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - (y - z)) * x;
	double tmp;
	if (x <= -6.5e-59) {
		tmp = t_1;
	} else if (x <= 1.4e-34) {
		tmp = (y - z) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (1.0 - (y - z)) * x
	tmp = 0
	if x <= -6.5e-59:
		tmp = t_1
	elif x <= 1.4e-34:
		tmp = (y - z) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 - Float64(y - z)) * x)
	tmp = 0.0
	if (x <= -6.5e-59)
		tmp = t_1;
	elseif (x <= 1.4e-34)
		tmp = Float64(Float64(y - z) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (1.0 - (y - z)) * x;
	tmp = 0.0;
	if (x <= -6.5e-59)
		tmp = t_1;
	elseif (x <= 1.4e-34)
		tmp = (y - z) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - N[(y - z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.5e-59], t$95$1, If[LessEqual[x, 1.4e-34], N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.50000000000000017e-59 or 1.39999999999999998e-34 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(1 - 1 \cdot \left(y - z\right)\right) \cdot x \]

      5. *-lft-identity N/A

         \[\leadsto \left(1 - \left(y - z\right)\right) \cdot x \]

      6. lower--.f64 N/A

         \[\leadsto \left(1 - \left(y - z\right)\right) \cdot x \]

      7. lift--.f64 86.8

         \[\leadsto \left(1 - \left(y - z\right)\right) \cdot x \]

   5. Applied rewrites 86.8%

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

   if -6.50000000000000017e-59 < x < 1.39999999999999998e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 85.3

         \[\leadsto \left(y - z\right) \cdot t \]

   5. Applied rewrites 85.3%

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

Alternative 4: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -1700000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)))
   (if (<= z -1700000.0) t_1 (if (<= z 2.35e+54) (fma (- t x) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double tmp;
	if (z <= -1700000.0) {
		tmp = t_1;
	} else if (z <= 2.35e+54) {
		tmp = fma((t - x), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - t) * z)
	tmp = 0.0
	if (z <= -1700000.0)
		tmp = t_1;
	elseif (z <= 2.35e+54)
		tmp = fma(Float64(t - x), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1700000.0], t$95$1, If[LessEqual[z, 2.35e+54], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e6 or 2.34999999999999996e54 < z

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift--.f64 N/A

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

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

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 77.7

         \[\leadsto \left(x - t\right) \cdot z \]

   7. Applied rewrites 77.7%

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

   if -1.7e6 < z < 2.34999999999999996e54

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(t - x\right) \cdot y + x \]

      3. lower-fma.f64 N/A

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

      4. lift--.f64 87.8

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

   5. Applied rewrites 87.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1700000:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ t_2 := x - z \cdot t\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-198}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)) (t_2 (- x (* z t))))
   (if (<= y -6.2e+15)
     t_1
     (if (<= y -8e-198)
       t_2
       (if (<= y 1.9e-148) (fma z x x) (if (<= y 1.3e-5) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double t_2 = x - (z * t);
	double tmp;
	if (y <= -6.2e+15) {
		tmp = t_1;
	} else if (y <= -8e-198) {
		tmp = t_2;
	} else if (y <= 1.9e-148) {
		tmp = fma(z, x, x);
	} else if (y <= 1.3e-5) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	t_2 = Float64(x - Float64(z * t))
	tmp = 0.0
	if (y <= -6.2e+15)
		tmp = t_1;
	elseif (y <= -8e-198)
		tmp = t_2;
	elseif (y <= 1.9e-148)
		tmp = fma(z, x, x);
	elseif (y <= 1.3e-5)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+15], t$95$1, If[LessEqual[y, -8e-198], t$95$2, If[LessEqual[y, 1.9e-148], N[(z * x + x), $MachinePrecision], If[LessEqual[y, 1.3e-5], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2e15 or 1.29999999999999992e-5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if -6.2e15 < y < -7.9999999999999993e-198 or 1.90000000000000007e-148 < y < 1.29999999999999992e-5

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift--.f64 N/A

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

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

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower--.f64 N/A

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

      3. lift--.f64 N/A

         \[\leadsto x - \left(t - x\right) \cdot z \]

      4. lift-*.f64 86.4

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto x - z \cdot t \]

      2. lower-*.f64 70.0

         \[\leadsto x - z \cdot t \]

   10. Applied rewrites 70.0%

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

   if -7.9999999999999993e-198 < y < 1.90000000000000007e-148

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift--.f64 N/A

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

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

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower--.f64 N/A

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

      3. lift--.f64 N/A

         \[\leadsto x - \left(t - x\right) \cdot z \]

      4. lift-*.f64 97.0

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

   7. Applied rewrites 97.0%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r* N/A

         \[\leadsto x - \left(-1 \cdot x\right) \cdot z \]

      2. mul-1-neg N/A

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

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

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

      4. +-commutative N/A

         \[\leadsto x \cdot z + x \]

      5. *-commutative N/A

         \[\leadsto z \cdot x + x \]

      6. lower-fma.f64 73.0

         \[\leadsto \mathsf{fma}\left(z, x, x\right) \]

   10. Applied rewrites 73.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-198}:\\ \;\;\;\;x - z \cdot t\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;x - z \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -0.06) t_1 (if (<= y 1.3e-5) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -0.06) {
		tmp = t_1;
	} else if (y <= 1.3e-5) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	tmp = 0.0
	if (y <= -0.06)
		tmp = t_1;
	elseif (y <= 1.3e-5)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -0.06], t$95$1, If[LessEqual[y, 1.3e-5], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.059999999999999998 or 1.29999999999999992e-5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -0.059999999999999998 < y < 1.29999999999999992e-5

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift--.f64 N/A

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

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

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower--.f64 N/A

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

      3. lift--.f64 N/A

         \[\leadsto x - \left(t - x\right) \cdot z \]

      4. lift-*.f64 93.2

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

   7. Applied rewrites 93.2%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r* N/A

         \[\leadsto x - \left(-1 \cdot x\right) \cdot z \]

      2. mul-1-neg N/A

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

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

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

      4. +-commutative N/A

         \[\leadsto x \cdot z + x \]

      5. *-commutative N/A

         \[\leadsto z \cdot x + x \]

      6. lower-fma.f64 65.7

         \[\leadsto \mathsf{fma}\left(z, x, x\right) \]

   10. Applied rewrites 65.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12000000:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{elif}\;y \leq 9000000000000:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -12000000.0)
   (* (- 1.0 y) x)
   (if (<= y 9000000000000.0) (fma z x x) (* (- x) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -12000000.0) {
		tmp = (1.0 - y) * x;
	} else if (y <= 9000000000000.0) {
		tmp = fma(z, x, x);
	} else {
		tmp = -x * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -12000000.0)
		tmp = Float64(Float64(1.0 - y) * x);
	elseif (y <= 9000000000000.0)
		tmp = fma(z, x, x);
	else
		tmp = Float64(Float64(-x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -12000000.0], N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 9000000000000.0], N[(z * x + x), $MachinePrecision], N[((-x) * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(1 - 1 \cdot \left(y - z\right)\right) \cdot x \]

      5. *-lft-identity N/A

         \[\leadsto \left(1 - \left(y - z\right)\right) \cdot x \]

      6. lower--.f64 N/A

         \[\leadsto \left(1 - \left(y - z\right)\right) \cdot x \]

      7. lift--.f64 52.0

         \[\leadsto \left(1 - \left(y - z\right)\right) \cdot x \]

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(1 - y\right) \cdot x \]
   7. Step-by-step derivation

      1. lower--.f64 47.9

         \[\leadsto \left(1 - y\right) \cdot x \]

   8. Applied rewrites 47.9%

      \[\leadsto \left(1 - y\right) \cdot x \]

   if -1.2e7 < y < 9e12

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift--.f64 N/A

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

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

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower--.f64 N/A

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

      3. lift--.f64 N/A

         \[\leadsto x - \left(t - x\right) \cdot z \]

      4. lift-*.f64 91.4

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

   7. Applied rewrites 91.4%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r* N/A

         \[\leadsto x - \left(-1 \cdot x\right) \cdot z \]

      2. mul-1-neg N/A

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

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

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

      4. +-commutative N/A

         \[\leadsto x \cdot z + x \]

      5. *-commutative N/A

         \[\leadsto z \cdot x + x \]

      6. lower-fma.f64 64.9

         \[\leadsto \mathsf{fma}\left(z, x, x\right) \]

   10. Applied rewrites 64.9%

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

   if 9e12 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 86.8

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot y \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 56.0

         \[\leadsto \left(-x\right) \cdot y \]

   8. Applied rewrites 56.0%

      \[\leadsto \left(-x\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12000000:\\ \;\;\;\;\left(1 - y\right) \cdot x\\ \mathbf{elif}\;y \leq 9000000000000:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot y\\ \mathbf{if}\;y \leq -950000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9000000000000:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x) y)))
   (if (<= y -950000000.0) t_1 (if (<= y 9000000000000.0) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -x * y;
	double tmp;
	if (y <= -950000000.0) {
		tmp = t_1;
	} else if (y <= 9000000000000.0) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-x) * y)
	tmp = 0.0
	if (y <= -950000000.0)
		tmp = t_1;
	elseif (y <= 9000000000000.0)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-x) * y), $MachinePrecision]}, If[LessEqual[y, -950000000.0], t$95$1, If[LessEqual[y, 9000000000000.0], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5e8 or 9e12 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot y \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.2

         \[\leadsto \left(-x\right) \cdot y \]

   8. Applied rewrites 51.2%

      \[\leadsto \left(-x\right) \cdot y \]

   if -9.5e8 < y < 9e12

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift--.f64 N/A

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

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

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

      20. lower--.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower--.f64 N/A

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

      3. lift--.f64 N/A

         \[\leadsto x - \left(t - x\right) \cdot z \]

      4. lift-*.f64 91.4

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

   7. Applied rewrites 91.4%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r* N/A

         \[\leadsto x - \left(-1 \cdot x\right) \cdot z \]

      2. mul-1-neg N/A

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

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

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

      4. +-commutative N/A

         \[\leadsto x \cdot z + x \]

      5. *-commutative N/A

         \[\leadsto z \cdot x + x \]

      6. lower-fma.f64 64.9

         \[\leadsto \mathsf{fma}\left(z, x, x\right) \]

   10. Applied rewrites 64.9%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -950000000:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{elif}\;y \leq 9000000000000:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-z\right) \cdot t\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(t, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- z) t)))
   (if (<= z -2.4e+102)
     t_1
     (if (<= z -1000000.0) (* z x) (if (<= z 6.1e+56) (fma t y x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -z * t;
	double tmp;
	if (z <= -2.4e+102) {
		tmp = t_1;
	} else if (z <= -1000000.0) {
		tmp = z * x;
	} else if (z <= 6.1e+56) {
		tmp = fma(t, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-z) * t)
	tmp = 0.0
	if (z <= -2.4e+102)
		tmp = t_1;
	elseif (z <= -1000000.0)
		tmp = Float64(z * x);
	elseif (z <= 6.1e+56)
		tmp = fma(t, y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-z) * t), $MachinePrecision]}, If[LessEqual[z, -2.4e+102], t$95$1, If[LessEqual[z, -1000000.0], N[(z * x), $MachinePrecision], If[LessEqual[z, 6.1e+56], N[(t * y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.39999999999999994e102 or 6.1000000000000001e56 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(-z\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 50.2%

      \[\leadsto \left(-z\right) \cdot t \]

   if -2.39999999999999994e102 < z < -1e6

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto z \cdot x \]

      2. lower-*.f64 52.2

         \[\leadsto z \cdot x \]

   8. Applied rewrites 52.2%

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

   if -1e6 < z < 6.1000000000000001e56

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(t - x\right) \cdot y + x \]

      3. lower-fma.f64 N/A

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

      4. lift--.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.9%

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

Alternative 10: 37.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot y\\ t_2 := \left(-z\right) \cdot t\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-198}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x) y)) (t_2 (* (- z) t)))
   (if (<= y -6.5e+15)
     t_1
     (if (<= y -8e-198)
       t_2
       (if (<= y 1.9e-148) (* z x) (if (<= y 1.3e-5) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -x * y;
	double t_2 = -z * t;
	double tmp;
	if (y <= -6.5e+15) {
		tmp = t_1;
	} else if (y <= -8e-198) {
		tmp = t_2;
	} else if (y <= 1.9e-148) {
		tmp = z * x;
	} else if (y <= 1.3e-5) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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
    t_2 = -z * t
    if (y <= (-6.5d+15)) then
        tmp = t_1
    else if (y <= (-8d-198)) then
        tmp = t_2
    else if (y <= 1.9d-148) then
        tmp = z * x
    else if (y <= 1.3d-5) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -x * y;
	double t_2 = -z * t;
	double tmp;
	if (y <= -6.5e+15) {
		tmp = t_1;
	} else if (y <= -8e-198) {
		tmp = t_2;
	} else if (y <= 1.9e-148) {
		tmp = z * x;
	} else if (y <= 1.3e-5) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -x * y
	t_2 = -z * t
	tmp = 0
	if y <= -6.5e+15:
		tmp = t_1
	elif y <= -8e-198:
		tmp = t_2
	elif y <= 1.9e-148:
		tmp = z * x
	elif y <= 1.3e-5:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-x) * y)
	t_2 = Float64(Float64(-z) * t)
	tmp = 0.0
	if (y <= -6.5e+15)
		tmp = t_1;
	elseif (y <= -8e-198)
		tmp = t_2;
	elseif (y <= 1.9e-148)
		tmp = Float64(z * x);
	elseif (y <= 1.3e-5)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -x * y;
	t_2 = -z * t;
	tmp = 0.0;
	if (y <= -6.5e+15)
		tmp = t_1;
	elseif (y <= -8e-198)
		tmp = t_2;
	elseif (y <= 1.9e-148)
		tmp = z * x;
	elseif (y <= 1.3e-5)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-x) * y), $MachinePrecision]}, Block[{t$95$2 = N[((-z) * t), $MachinePrecision]}, If[LessEqual[y, -6.5e+15], t$95$1, If[LessEqual[y, -8e-198], t$95$2, If[LessEqual[y, 1.9e-148], N[(z * x), $MachinePrecision], If[LessEqual[y, 1.3e-5], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5e15 or 1.29999999999999992e-5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot y \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.0

         \[\leadsto \left(-x\right) \cdot y \]

   8. Applied rewrites 51.0%

      \[\leadsto \left(-x\right) \cdot y \]

   if -6.5e15 < y < -7.9999999999999993e-198 or 1.90000000000000007e-148 < y < 1.29999999999999992e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(-z\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 41.1%

      \[\leadsto \left(-z\right) \cdot t \]

   if -7.9999999999999993e-198 < y < 1.90000000000000007e-148

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto z \cdot x \]

      2. lower-*.f64 45.1

         \[\leadsto z \cdot x \]

   8. Applied rewrites 45.1%

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

Alternative 11: 36.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot y\\ \mathbf{if}\;y \leq -42000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9000000000000:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x) y)))
   (if (<= y -42000000.0) t_1 (if (<= y 9000000000000.0) (* z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -x * y;
	double tmp;
	if (y <= -42000000.0) {
		tmp = t_1;
	} else if (y <= 9000000000000.0) {
		tmp = z * x;
	} else {
		tmp = t_1;
	}
	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
    if (y <= (-42000000.0d0)) then
        tmp = t_1
    else if (y <= 9000000000000.0d0) then
        tmp = z * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -x * y;
	double tmp;
	if (y <= -42000000.0) {
		tmp = t_1;
	} else if (y <= 9000000000000.0) {
		tmp = z * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -x * y
	tmp = 0
	if y <= -42000000.0:
		tmp = t_1
	elif y <= 9000000000000.0:
		tmp = z * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-x) * y)
	tmp = 0.0
	if (y <= -42000000.0)
		tmp = t_1;
	elseif (y <= 9000000000000.0)
		tmp = Float64(z * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -x * y;
	tmp = 0.0;
	if (y <= -42000000.0)
		tmp = t_1;
	elseif (y <= 9000000000000.0)
		tmp = z * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-x) * y), $MachinePrecision]}, If[LessEqual[y, -42000000.0], t$95$1, If[LessEqual[y, 9000000000000.0], N[(z * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e7 or 9e12 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot y \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.2

         \[\leadsto \left(-x\right) \cdot y \]

   8. Applied rewrites 51.2%

      \[\leadsto \left(-x\right) \cdot y \]

   if -4.2e7 < y < 9e12

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto z \cdot x \]

      2. lower-*.f64 36.0

         \[\leadsto z \cdot x \]

   8. Applied rewrites 36.0%

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

Alternative 12: 36.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-159}:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1000000.0)
   (* z x)
   (if (<= z -3.7e-159) (* y t) (if (<= z 6e-15) x (* z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1000000.0) {
		tmp = z * x;
	} else if (z <= -3.7e-159) {
		tmp = y * t;
	} else if (z <= 6e-15) {
		tmp = x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1000000.0d0)) then
        tmp = z * x
    else if (z <= (-3.7d-159)) then
        tmp = y * t
    else if (z <= 6d-15) then
        tmp = x
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1000000.0) {
		tmp = z * x;
	} else if (z <= -3.7e-159) {
		tmp = y * t;
	} else if (z <= 6e-15) {
		tmp = x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1000000.0:
		tmp = z * x
	elif z <= -3.7e-159:
		tmp = y * t
	elif z <= 6e-15:
		tmp = x
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1000000.0)
		tmp = Float64(z * x);
	elseif (z <= -3.7e-159)
		tmp = Float64(y * t);
	elseif (z <= 6e-15)
		tmp = x;
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1000000.0)
		tmp = z * x;
	elseif (z <= -3.7e-159)
		tmp = y * t;
	elseif (z <= 6e-15)
		tmp = x;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1000000.0], N[(z * x), $MachinePrecision], If[LessEqual[z, -3.7e-159], N[(y * t), $MachinePrecision], If[LessEqual[z, 6e-15], x, N[(z * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e6 or 6e-15 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 73.4

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto z \cdot x \]

      2. lower-*.f64 41.2

         \[\leadsto z \cdot x \]

   8. Applied rewrites 41.2%

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

   if -1e6 < z < -3.6999999999999999e-159

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 51.5

         \[\leadsto \left(y - z\right) \cdot t \]

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto y \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 42.5%

      \[\leadsto y \cdot t \]

   if -3.6999999999999999e-159 < z < 6e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(t - x\right) \cdot y + x \]

      3. lower-fma.f64 N/A

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

      4. lift--.f64 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \]
   7. Step-by-step derivation

   8. Applied rewrites 41.5%

      \[\leadsto x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 35.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.0) (* z x) (if (<= z 6e-15) x (* z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.0) {
		tmp = z * x;
	} else if (z <= 6e-15) {
		tmp = x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = z * x
    else if (z <= 6d-15) then
        tmp = x
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.0) {
		tmp = z * x;
	} else if (z <= 6e-15) {
		tmp = x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.0:
		tmp = z * x
	elif z <= 6e-15:
		tmp = x
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(z * x);
	elseif (z <= 6e-15)
		tmp = x;
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = z * x;
	elseif (z <= 6e-15)
		tmp = x;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.0], N[(z * x), $MachinePrecision], If[LessEqual[z, 6e-15], x, N[(z * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 6e-15 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 73.4

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto z \cdot x \]

      2. lower-*.f64 41.2

         \[\leadsto z \cdot x \]

   8. Applied rewrites 41.2%

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

   if -1 < z < 6e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(t - x\right) \cdot y + x \]

      3. lower-fma.f64 N/A

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

      4. lift--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \]
   7. Step-by-step derivation

   8. Applied rewrites 35.2%

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

Alternative 14: 17.3% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

      \[\leadsto \left(t - x\right) \cdot y + x \]

   3. lower-fma.f64 N/A

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

   4. lift--.f64 60.5

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

5. Applied rewrites 60.5%

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

6. Taylor expanded in y around 0

   \[\leadsto x \]
7. Step-by-step derivation

8. Applied rewrites 17.6%

   \[\leadsto x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64

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

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