Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

Alternatives: 12

Speedup: 1.0×

• 34.4% 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: 37.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y - z \leq -2 \cdot 10^{+142}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y - z \leq -500:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;y - z \leq 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;y - z \leq 2 \cdot 10^{+199}:\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- y z) -2e+142)
   (* t y)
   (if (<= (- y z) -500.0)
     (* z x)
     (if (<= (- y z) 1e-6) x (if (<= (- y z) 2e+199) (* t y) (* z x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y - z) <= -2e+142) {
		tmp = t * y;
	} else if ((y - z) <= -500.0) {
		tmp = z * x;
	} else if ((y - z) <= 1e-6) {
		tmp = x;
	} else if ((y - z) <= 2e+199) {
		tmp = t * y;
	} 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 ((y - z) <= (-2d+142)) then
        tmp = t * y
    else if ((y - z) <= (-500.0d0)) then
        tmp = z * x
    else if ((y - z) <= 1d-6) then
        tmp = x
    else if ((y - z) <= 2d+199) then
        tmp = t * y
    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 ((y - z) <= -2e+142) {
		tmp = t * y;
	} else if ((y - z) <= -500.0) {
		tmp = z * x;
	} else if ((y - z) <= 1e-6) {
		tmp = x;
	} else if ((y - z) <= 2e+199) {
		tmp = t * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y - z) <= -2e+142:
		tmp = t * y
	elif (y - z) <= -500.0:
		tmp = z * x
	elif (y - z) <= 1e-6:
		tmp = x
	elif (y - z) <= 2e+199:
		tmp = t * y
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y - z) <= -2e+142)
		tmp = Float64(t * y);
	elseif (Float64(y - z) <= -500.0)
		tmp = Float64(z * x);
	elseif (Float64(y - z) <= 1e-6)
		tmp = x;
	elseif (Float64(y - z) <= 2e+199)
		tmp = Float64(t * y);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y - z) <= -2e+142)
		tmp = t * y;
	elseif ((y - z) <= -500.0)
		tmp = z * x;
	elseif ((y - z) <= 1e-6)
		tmp = x;
	elseif ((y - z) <= 2e+199)
		tmp = t * y;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y - z), $MachinePrecision], -2e+142], N[(t * y), $MachinePrecision], If[LessEqual[N[(y - z), $MachinePrecision], -500.0], N[(z * x), $MachinePrecision], If[LessEqual[N[(y - z), $MachinePrecision], 1e-6], x, If[LessEqual[N[(y - z), $MachinePrecision], 2e+199], N[(t * y), $MachinePrecision], N[(z * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 y z) < -2.0000000000000001e142 or 9.99999999999999955e-7 < (-.f64 y z) < 2.00000000000000019e199

   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 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.0%

      \[\leadsto t \cdot y \]

   if -2.0000000000000001e142 < (-.f64 y z) < -500 or 2.00000000000000019e199 < (-.f64 y z)

   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 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 41.2%

      \[\leadsto z \cdot x \]

   if -500 < (-.f64 y z) < 9.99999999999999955e-7

   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 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.4%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - z \leq -2 \cdot 10^{+142}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y - z \leq -500:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;y - z \leq 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;y - z \leq 2 \cdot 10^{+199}:\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot t\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+200}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0017:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) t)))
   (if (<= z -1.9e+200)
     (* z x)
     (if (<= z -6.8e+20)
       t_1
       (if (<= z 0.0017)
         (fma (- t x) y x)
         (if (<= z 5.5e+93) (fma x z x) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * t;
	double tmp;
	if (z <= -1.9e+200) {
		tmp = z * x;
	} else if (z <= -6.8e+20) {
		tmp = t_1;
	} else if (z <= 0.0017) {
		tmp = fma((t - x), y, x);
	} else if (z <= 5.5e+93) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * t)
	tmp = 0.0
	if (z <= -1.9e+200)
		tmp = Float64(z * x);
	elseif (z <= -6.8e+20)
		tmp = t_1;
	elseif (z <= 0.0017)
		tmp = fma(Float64(t - x), y, x);
	elseif (z <= 5.5e+93)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -1.9e+200], N[(z * x), $MachinePrecision], If[LessEqual[z, -6.8e+20], t$95$1, If[LessEqual[z, 0.0017], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 5.5e+93], N[(x * z + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.89999999999999991e200

   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 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 71.8%

      \[\leadsto z \cdot x \]

   if -1.89999999999999991e200 < z < -6.8e20 or 5.5000000000000003e93 < z

   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 58.3

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

   5. Applied rewrites 58.3%

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

   if -6.8e20 < z < 0.00169999999999999991

   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 92.6

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

   5. Applied rewrites 92.6%

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

   if 0.00169999999999999991 < z < 5.5000000000000003e93

   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

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

      4. lift-*.f64 N/A

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

      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. distribute-rgt-out N/A

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

      11. associate-*r* N/A

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

      12. associate-+l+ N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift--.f64 74.1

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 60.8

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

   10. Applied rewrites 60.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+200}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{+20}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{elif}\;z \leq 0.0017:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+35}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -1.7e-11)
     t_1
     (if (<= y 1.6e-61) (fma x z x) (if (<= y 6.2e+35) (* (- y z) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -1.7e-11) {
		tmp = t_1;
	} else if (y <= 1.6e-61) {
		tmp = fma(x, z, x);
	} else if (y <= 6.2e+35) {
		tmp = (y - z) * t;
	} 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 <= -1.7e-11)
		tmp = t_1;
	elseif (y <= 1.6e-61)
		tmp = fma(x, z, x);
	elseif (y <= 6.2e+35)
		tmp = Float64(Float64(y - z) * t);
	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, -1.7e-11], t$95$1, If[LessEqual[y, 1.6e-61], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 6.2e+35], N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999999e-11 or 6.19999999999999973e35 < 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 79.4

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

   5. Applied rewrites 79.4%

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

   if -1.6999999999999999e-11 < y < 1.6000000000000001e-61

   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

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

      4. lift-*.f64 N/A

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

      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. distribute-rgt-out N/A

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

      11. associate-*r* N/A

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

      12. associate-+l+ N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift--.f64 93.0

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.1

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

   10. Applied rewrites 66.1%

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

   if 1.6000000000000001e-61 < y < 6.19999999999999973e35

   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 83.7

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

   5. Applied rewrites 83.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-11}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+35}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.46 \cdot 10^{+200}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -12000000:\\ \;\;\;\;\left(-t\right) \cdot z\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(t, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.46e+200)
   (* z x)
   (if (<= z -12000000.0)
     (* (- t) z)
     (if (<= z 2.8e-11) (fma t y x) (fma x z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.46e+200) {
		tmp = z * x;
	} else if (z <= -12000000.0) {
		tmp = -t * z;
	} else if (z <= 2.8e-11) {
		tmp = fma(t, y, x);
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.46e+200)
		tmp = Float64(z * x);
	elseif (z <= -12000000.0)
		tmp = Float64(Float64(-t) * z);
	elseif (z <= 2.8e-11)
		tmp = fma(t, y, x);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.46e+200], N[(z * x), $MachinePrecision], If[LessEqual[z, -12000000.0], N[((-t) * z), $MachinePrecision], If[LessEqual[z, 2.8e-11], N[(t * y + x), $MachinePrecision], N[(x * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.46e200

   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 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 71.8%

      \[\leadsto z \cdot x \]

   if -1.46e200 < z < -1.2e7

   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

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

      4. lift-*.f64 N/A

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

      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. distribute-rgt-out N/A

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

      11. associate-*r* N/A

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

      12. associate-+l+ N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lift--.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift--.f64 62.5

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

   7. Applied rewrites 62.5%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 38.2

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

   10. Applied rewrites 38.2%

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

   if -1.2e7 < z < 2.8e-11

   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 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.2%

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

   if 2.8e-11 < z

   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

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

      4. lift-*.f64 N/A

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

      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. distribute-rgt-out N/A

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

      11. associate-*r* N/A

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

      12. associate-+l+ N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift--.f64 77.0

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 46.0

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

   10. Applied rewrites 46.0%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.46 \cdot 10^{+200}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -12000000:\\ \;\;\;\;\left(-t\right) \cdot z\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(t, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+20}:\\ \;\;\;\;\left(-z\right) \cdot \left(t - x\right)\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.5e+20)
   (* (- z) (- t x))
   (if (<= z 1.36e-8) (fma (- t x) y x) (fma (- z) (- t x) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.5e+20) {
		tmp = -z * (t - x);
	} else if (z <= 1.36e-8) {
		tmp = fma((t - x), y, x);
	} else {
		tmp = fma(-z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.5e+20)
		tmp = Float64(Float64(-z) * Float64(t - x));
	elseif (z <= 1.36e-8)
		tmp = fma(Float64(t - x), y, x);
	else
		tmp = fma(Float64(-z), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.5e+20], N[((-z) * N[(t - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.36e-8], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], N[((-z) * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.5e20

   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 76.7

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

   5. Applied rewrites 76.7%

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

   if -8.5e20 < z < 1.3599999999999999e-8

   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 93.3

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

   5. Applied rewrites 93.3%

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

   if 1.3599999999999999e-8 < z

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 78.2

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

   5. Applied rewrites 78.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+20}:\\ \;\;\;\;\left(-z\right) \cdot \left(t - x\right)\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+20} \lor \neg \left(z \leq 1.36 \cdot 10^{-8}\right):\\ \;\;\;\;\left(-z\right) \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.5e+20) (not (<= z 1.36e-8)))
   (* (- z) (- t x))
   (fma (- t x) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.5e+20) || !(z <= 1.36e-8)) {
		tmp = -z * (t - x);
	} else {
		tmp = fma((t - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.5e+20) || !(z <= 1.36e-8))
		tmp = Float64(Float64(-z) * Float64(t - x));
	else
		tmp = fma(Float64(t - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.5e+20], N[Not[LessEqual[z, 1.36e-8]], $MachinePrecision]], N[((-z) * N[(t - x), $MachinePrecision]), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e20 or 1.3599999999999999e-8 < 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 77.0

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

   5. Applied rewrites 77.0%

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

   if -8.5e20 < z < 1.3599999999999999e-8

   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 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+20} \lor \neg \left(z \leq 1.36 \cdot 10^{-8}\right):\\ \;\;\;\;\left(-z\right) \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-11} \lor \neg \left(y \leq 9.8 \cdot 10^{-50}\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.7e-11) (not (<= y 9.8e-50))) (* (- t x) y) (fma x z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.7e-11) || !(y <= 9.8e-50)) {
		tmp = (t - x) * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.7e-11) || !(y <= 9.8e-50))
		tmp = Float64(Float64(t - x) * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.7e-11], N[Not[LessEqual[y, 9.8e-50]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6999999999999999e-11 or 9.7999999999999997e-50 < 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 76.4

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

   5. Applied rewrites 76.4%

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

   if -1.6999999999999999e-11 < y < 9.7999999999999997e-50

   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

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

      4. lift-*.f64 N/A

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

      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. distribute-rgt-out N/A

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

      11. associate-*r* N/A

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

      12. associate-+l+ N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift--.f64 93.0

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.1

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

   10. Applied rewrites 66.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-11} \lor \neg \left(y \leq 9.8 \cdot 10^{-50}\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -180000000 \lor \neg \left(z \leq 2.8 \cdot 10^{-11}\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -180000000.0) (not (<= z 2.8e-11))) (* z x) (fma t y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -180000000.0) || !(z <= 2.8e-11)) {
		tmp = z * x;
	} else {
		tmp = fma(t, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -180000000.0) || !(z <= 2.8e-11))
		tmp = Float64(z * x);
	else
		tmp = fma(t, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -180000000.0], N[Not[LessEqual[z, 2.8e-11]], $MachinePrecision]], N[(z * x), $MachinePrecision], N[(t * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e8 or 2.8e-11 < z

   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 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 44.0%

      \[\leadsto z \cdot x \]

   if -1.8e8 < z < 2.8e-11

   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 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -180000000 \lor \neg \left(z \leq 2.8 \cdot 10^{-11}\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -180000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(t, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -180000000.0) (* z x) (if (<= z 2.8e-11) (fma t y x) (fma x z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -180000000.0) {
		tmp = z * x;
	} else if (z <= 2.8e-11) {
		tmp = fma(t, y, x);
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -180000000.0)
		tmp = Float64(z * x);
	elseif (z <= 2.8e-11)
		tmp = fma(t, y, x);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -180000000.0], N[(z * x), $MachinePrecision], If[LessEqual[z, 2.8e-11], N[(t * y + x), $MachinePrecision], N[(x * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e8

   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 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 42.7%

      \[\leadsto z \cdot x \]

   if -1.8e8 < z < 2.8e-11

   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 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.2%

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

   if 2.8e-11 < z

   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

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

      4. lift-*.f64 N/A

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

      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. distribute-rgt-out N/A

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

      11. associate-*r* N/A

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

      12. associate-+l+ N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift--.f64 77.0

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 46.0

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

   10. Applied rewrites 46.0%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -180000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(t, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-6} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.4e-6) (not (<= z 1.0))) (* z x) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e-6) || !(z <= 1.0)) {
		tmp = z * x;
	} else {
		tmp = 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 <= (-3.4d-6)) .or. (.not. (z <= 1.0d0))) then
        tmp = z * x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e-6) || !(z <= 1.0)) {
		tmp = z * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.4e-6) or not (z <= 1.0):
		tmp = z * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.4e-6) || !(z <= 1.0))
		tmp = Float64(z * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.4e-6) || ~((z <= 1.0)))
		tmp = z * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.4e-6], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(z * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -3.40000000000000006e-6 or 1 < z

   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 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 44.6%

      \[\leadsto z \cdot x \]

   if -3.40000000000000006e-6 < z < 1

   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 91.8

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 32.1%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-6} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 17.9% accurate, 15.0× 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 63.5

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

5. Applied rewrites 63.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 18.3%

   \[\leadsto x \]

9. Final simplification 18.3%

   \[\leadsto x \]
10. Add Preprocessing

Developer Target 1: 96.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025084 
(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))))