Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 2.8s

Alternatives: 13

Speedup: 1.0×

• 34.7% 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

Alternative 2: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.6:\\ \;\;\;\;x - \left(t - x\right) \cdot z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, z, z \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.6)
   (- x (* (- t x) z))
   (if (<= z 3.6e+49) (fma (- t x) y x) (fma (- t) z (* z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.6) {
		tmp = x - ((t - x) * z);
	} else if (z <= 3.6e+49) {
		tmp = fma((t - x), y, x);
	} else {
		tmp = fma(-t, z, (z * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.6)
		tmp = Float64(x - Float64(Float64(t - x) * z));
	elseif (z <= 3.6e+49)
		tmp = fma(Float64(t - x), y, x);
	else
		tmp = fma(Float64(-t), z, Float64(z * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.6], N[(x - N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e+49], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], N[((-t) * z + N[(z * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.599999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      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 76.9

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

   4. Applied rewrites 76.9%

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

   if -0.599999999999999978 < z < 3.59999999999999996e49

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
   3. 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 87.5

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

   4. Applied rewrites 87.5%

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

   if 3.59999999999999996e49 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.3

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

   7. Applied rewrites 79.3%

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

Alternative 3: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.6:\\ \;\;\;\;x - \left(t - x\right) \cdot z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.6)
   (- x (* (- t x) z))
   (if (<= z 3.6e+49) (fma (- t x) y x) (* (- x t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.6) {
		tmp = x - ((t - x) * z);
	} else if (z <= 3.6e+49) {
		tmp = fma((t - x), y, x);
	} else {
		tmp = (x - t) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.6)
		tmp = Float64(x - Float64(Float64(t - x) * z));
	elseif (z <= 3.6e+49)
		tmp = fma(Float64(t - x), y, x);
	else
		tmp = Float64(Float64(x - t) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.6], N[(x - N[(N[(t - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e+49], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.599999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      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 76.9

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

   4. Applied rewrites 76.9%

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

   if -0.599999999999999978 < z < 3.59999999999999996e49

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
   3. 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 87.5

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

   4. Applied rewrites 87.5%

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

   if 3.59999999999999996e49 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. negate-sub2 N/A

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

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

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 93.1

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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in z around inf

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

      1. negate-sub2 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub2 N/A

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

      7. lift--.f64 81.9

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

   7. Applied rewrites 81.9%

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

Alternative 4: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -2150000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)))
   (if (<= z -2150000000.0) t_1 (if (<= z 3.6e+49) (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 <= -2150000000.0) {
		tmp = t_1;
	} else if (z <= 3.6e+49) {
		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 <= -2150000000.0)
		tmp = t_1;
	elseif (z <= 3.6e+49)
		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, -2150000000.0], t$95$1, If[LessEqual[z, 3.6e+49], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.15e9 or 3.59999999999999996e49 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. negate-sub2 N/A

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

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

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in z around inf

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

      1. negate-sub2 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub2 N/A

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

      7. lift--.f64 79.5

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

   7. Applied rewrites 79.5%

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

   if -2.15e9 < z < 3.59999999999999996e49

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
   3. 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 86.9

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

   4. Applied rewrites 86.9%

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

Alternative 5: 68.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -1.35e+18) t_1 (if (<= y 5e+51) (* (- x t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -1.35e+18) {
		tmp = t_1;
	} else if (y <= 5e+51) {
		tmp = (x - t) * z;
	} 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 = (t - x) * y
    if (y <= (-1.35d+18)) then
        tmp = t_1
    else if (y <= 5d+51) then
        tmp = (x - t) * z
    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 = (t - x) * y;
	double tmp;
	if (y <= -1.35e+18) {
		tmp = t_1;
	} else if (y <= 5e+51) {
		tmp = (x - t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t - x) * y
	tmp = 0
	if y <= -1.35e+18:
		tmp = t_1
	elif y <= 5e+51:
		tmp = (x - t) * z
	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 <= -1.35e+18)
		tmp = t_1;
	elseif (y <= 5e+51)
		tmp = Float64(Float64(x - t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t - x) * y;
	tmp = 0.0;
	if (y <= -1.35e+18)
		tmp = t_1;
	elseif (y <= 5e+51)
		tmp = (x - t) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.35e+18], t$95$1, If[LessEqual[y, 5e+51], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e18 or 5e51 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   3. 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 81.9

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

   4. Applied rewrites 81.9%

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

   if -1.35e18 < y < 5e51

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. negate-sub2 N/A

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

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

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. negate-sub2 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub2 N/A

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

      7. lift--.f64 57.6

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

   7. Applied rewrites 57.6%

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

Alternative 6: 63.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ t_2 := \left(1 - y\right) \cdot x\\ \mathbf{if}\;z \leq -1.25:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-271}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-95}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+48}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)) (t_2 (* (- 1.0 y) x)))
   (if (<= z -1.25)
     t_1
     (if (<= z 3.8e-271)
       t_2
       (if (<= z 6e-95) (* t y) (if (<= z 9.2e+48) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double t_2 = (1.0 - y) * x;
	double tmp;
	if (z <= -1.25) {
		tmp = t_1;
	} else if (z <= 3.8e-271) {
		tmp = t_2;
	} else if (z <= 6e-95) {
		tmp = t * y;
	} else if (z <= 9.2e+48) {
		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 - t) * z
    t_2 = (1.0d0 - y) * x
    if (z <= (-1.25d0)) then
        tmp = t_1
    else if (z <= 3.8d-271) then
        tmp = t_2
    else if (z <= 6d-95) then
        tmp = t * y
    else if (z <= 9.2d+48) 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 - t) * z;
	double t_2 = (1.0 - y) * x;
	double tmp;
	if (z <= -1.25) {
		tmp = t_1;
	} else if (z <= 3.8e-271) {
		tmp = t_2;
	} else if (z <= 6e-95) {
		tmp = t * y;
	} else if (z <= 9.2e+48) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - t) * z
	t_2 = (1.0 - y) * x
	tmp = 0
	if z <= -1.25:
		tmp = t_1
	elif z <= 3.8e-271:
		tmp = t_2
	elif z <= 6e-95:
		tmp = t * y
	elif z <= 9.2e+48:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - t) * z)
	t_2 = Float64(Float64(1.0 - y) * x)
	tmp = 0.0
	if (z <= -1.25)
		tmp = t_1;
	elseif (z <= 3.8e-271)
		tmp = t_2;
	elseif (z <= 6e-95)
		tmp = Float64(t * y);
	elseif (z <= 9.2e+48)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - t) * z;
	t_2 = (1.0 - y) * x;
	tmp = 0.0;
	if (z <= -1.25)
		tmp = t_1;
	elseif (z <= 3.8e-271)
		tmp = t_2;
	elseif (z <= 6e-95)
		tmp = t * y;
	elseif (z <= 9.2e+48)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.25], t$95$1, If[LessEqual[z, 3.8e-271], t$95$2, If[LessEqual[z, 6e-95], N[(t * y), $MachinePrecision], If[LessEqual[z, 9.2e+48], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25 or 9.2000000000000001e48 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. negate-sub2 N/A

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

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

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in z around inf

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

      1. negate-sub2 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub2 N/A

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

      7. lift--.f64 78.9

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

   7. Applied rewrites 78.9%

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

   if -1.25 < z < 3.8000000000000001e-271 or 6e-95 < z < 9.2000000000000001e48

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 58.4

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 55.3

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

   7. Applied rewrites 55.3%

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

   if 3.8000000000000001e-271 < z < 6e-95

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   3. 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 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 37.4%

      \[\leadsto t \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-y\right) \cdot x\\ t_2 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -400:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-90}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-56}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y) x)) (t_2 (* (- x t) z)))
   (if (<= z -400.0)
     t_2
     (if (<= z -2.2e-167)
       (fma z x x)
       (if (<= z -3e-290)
         t_1
         (if (<= z 4e-90) (* t y) (if (<= z 1.36e-56) t_1 t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -y * x;
	double t_2 = (x - t) * z;
	double tmp;
	if (z <= -400.0) {
		tmp = t_2;
	} else if (z <= -2.2e-167) {
		tmp = fma(z, x, x);
	} else if (z <= -3e-290) {
		tmp = t_1;
	} else if (z <= 4e-90) {
		tmp = t * y;
	} else if (z <= 1.36e-56) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-y) * x)
	t_2 = Float64(Float64(x - t) * z)
	tmp = 0.0
	if (z <= -400.0)
		tmp = t_2;
	elseif (z <= -2.2e-167)
		tmp = fma(z, x, x);
	elseif (z <= -3e-290)
		tmp = t_1;
	elseif (z <= 4e-90)
		tmp = Float64(t * y);
	elseif (z <= 1.36e-56)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -400.0], t$95$2, If[LessEqual[z, -2.2e-167], N[(z * x + x), $MachinePrecision], If[LessEqual[z, -3e-290], t$95$1, If[LessEqual[z, 4e-90], N[(t * y), $MachinePrecision], If[LessEqual[z, 1.36e-56], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -400 or 1.36000000000000006e-56 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. negate-sub2 N/A

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

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

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in z around inf

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

      1. negate-sub2 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub2 N/A

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

      7. lift--.f64 72.0

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

   7. Applied rewrites 72.0%

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

   if -400 < z < -2.2e-167

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{-1 \cdot \left(x \cdot z\right)} \]
   6. 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 29.5

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

   7. Applied rewrites 29.5%

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

   if -2.2e-167 < z < -2.99999999999999992e-290 or 3.99999999999999998e-90 < z < 1.36000000000000006e-56

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.7

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

   7. Applied rewrites 27.7%

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

   if -2.99999999999999992e-290 < z < 3.99999999999999998e-90

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   3. 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 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 38.0%

      \[\leadsto t \cdot y \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 50.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+174}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq -9 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y) x)))
   (if (<= y -2.9e+174)
     (* t y)
     (if (<= y -9e+16) t_1 (if (<= y 3.5e+51) (fma z x x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -y * x;
	double tmp;
	if (y <= -2.9e+174) {
		tmp = t * y;
	} else if (y <= -9e+16) {
		tmp = t_1;
	} else if (y <= 3.5e+51) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (y <= -2.9e+174)
		tmp = Float64(t * y);
	elseif (y <= -9e+16)
		tmp = t_1;
	elseif (y <= 3.5e+51)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[y, -2.9e+174], N[(t * y), $MachinePrecision], If[LessEqual[y, -9e+16], t$95$1, If[LessEqual[y, 3.5e+51], N[(z * x + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9e174

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   3. 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 92.5

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 50.3%

      \[\leadsto t \cdot y \]

   if -2.9e174 < y < -9e16 or 3.5e51 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 52.8

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 41.6

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

   7. Applied rewrites 41.6%

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

   if -9e16 < y < 3.5e51

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 59.1

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{-1 \cdot \left(x \cdot z\right)} \]
   6. 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 55.7

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

   7. Applied rewrites 55.7%

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

Alternative 9: 38.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+125}:\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{elif}\;z \leq -2800000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-90}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y) x)))
   (if (<= z -3.8e+125)
     (* (- z) t)
     (if (<= z -2800000000.0)
       (* z x)
       (if (<= z -3e-290)
         t_1
         (if (<= z 4e-90) (* t y) (if (<= z 4.2e+49) t_1 (* z x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -y * x;
	double tmp;
	if (z <= -3.8e+125) {
		tmp = -z * t;
	} else if (z <= -2800000000.0) {
		tmp = z * x;
	} else if (z <= -3e-290) {
		tmp = t_1;
	} else if (z <= 4e-90) {
		tmp = t * y;
	} else if (z <= 4.2e+49) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = -y * x
    if (z <= (-3.8d+125)) then
        tmp = -z * t
    else if (z <= (-2800000000.0d0)) then
        tmp = z * x
    else if (z <= (-3d-290)) then
        tmp = t_1
    else if (z <= 4d-90) then
        tmp = t * y
    else if (z <= 4.2d+49) then
        tmp = t_1
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -y * x;
	double tmp;
	if (z <= -3.8e+125) {
		tmp = -z * t;
	} else if (z <= -2800000000.0) {
		tmp = z * x;
	} else if (z <= -3e-290) {
		tmp = t_1;
	} else if (z <= 4e-90) {
		tmp = t * y;
	} else if (z <= 4.2e+49) {
		tmp = t_1;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -y * x
	tmp = 0
	if z <= -3.8e+125:
		tmp = -z * t
	elif z <= -2800000000.0:
		tmp = z * x
	elif z <= -3e-290:
		tmp = t_1
	elif z <= 4e-90:
		tmp = t * y
	elif z <= 4.2e+49:
		tmp = t_1
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (z <= -3.8e+125)
		tmp = Float64(Float64(-z) * t);
	elseif (z <= -2800000000.0)
		tmp = Float64(z * x);
	elseif (z <= -3e-290)
		tmp = t_1;
	elseif (z <= 4e-90)
		tmp = Float64(t * y);
	elseif (z <= 4.2e+49)
		tmp = t_1;
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -y * x;
	tmp = 0.0;
	if (z <= -3.8e+125)
		tmp = -z * t;
	elseif (z <= -2800000000.0)
		tmp = z * x;
	elseif (z <= -3e-290)
		tmp = t_1;
	elseif (z <= 4e-90)
		tmp = t * y;
	elseif (z <= 4.2e+49)
		tmp = t_1;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[z, -3.8e+125], N[((-z) * t), $MachinePrecision], If[LessEqual[z, -2800000000.0], N[(z * x), $MachinePrecision], If[LessEqual[z, -3e-290], t$95$1, If[LessEqual[z, 4e-90], N[(t * y), $MachinePrecision], If[LessEqual[z, 4.2e+49], t$95$1, N[(z * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.80000000000000002e125

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 88.2

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 47.3%

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

   if -3.80000000000000002e125 < z < -2.8e9 or 4.20000000000000022e49 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 40.6%

      \[\leadsto z \cdot x \]

   if -2.8e9 < z < -2.99999999999999992e-290 or 3.99999999999999998e-90 < z < 4.20000000000000022e49

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 57.5

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.9

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

   7. Applied rewrites 27.9%

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

   if -2.99999999999999992e-290 < z < 3.99999999999999998e-90

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   3. 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 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 38.0%

      \[\leadsto t \cdot y \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 36.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;z \leq -2800000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-90}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y) x)))
   (if (<= z -2800000000.0)
     (* z x)
     (if (<= z -3e-290)
       t_1
       (if (<= z 4e-90) (* t y) (if (<= z 4.2e+49) t_1 (* z x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -y * x;
	double tmp;
	if (z <= -2800000000.0) {
		tmp = z * x;
	} else if (z <= -3e-290) {
		tmp = t_1;
	} else if (z <= 4e-90) {
		tmp = t * y;
	} else if (z <= 4.2e+49) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = -y * x
    if (z <= (-2800000000.0d0)) then
        tmp = z * x
    else if (z <= (-3d-290)) then
        tmp = t_1
    else if (z <= 4d-90) then
        tmp = t * y
    else if (z <= 4.2d+49) then
        tmp = t_1
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -y * x;
	double tmp;
	if (z <= -2800000000.0) {
		tmp = z * x;
	} else if (z <= -3e-290) {
		tmp = t_1;
	} else if (z <= 4e-90) {
		tmp = t * y;
	} else if (z <= 4.2e+49) {
		tmp = t_1;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -y * x
	tmp = 0
	if z <= -2800000000.0:
		tmp = z * x
	elif z <= -3e-290:
		tmp = t_1
	elif z <= 4e-90:
		tmp = t * y
	elif z <= 4.2e+49:
		tmp = t_1
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (z <= -2800000000.0)
		tmp = Float64(z * x);
	elseif (z <= -3e-290)
		tmp = t_1;
	elseif (z <= 4e-90)
		tmp = Float64(t * y);
	elseif (z <= 4.2e+49)
		tmp = t_1;
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -y * x;
	tmp = 0.0;
	if (z <= -2800000000.0)
		tmp = z * x;
	elseif (z <= -3e-290)
		tmp = t_1;
	elseif (z <= 4e-90)
		tmp = t * y;
	elseif (z <= 4.2e+49)
		tmp = t_1;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[z, -2800000000.0], N[(z * x), $MachinePrecision], If[LessEqual[z, -3e-290], t$95$1, If[LessEqual[z, 4e-90], N[(t * y), $MachinePrecision], If[LessEqual[z, 4.2e+49], t$95$1, N[(z * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8e9 or 4.20000000000000022e49 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 43.1%

      \[\leadsto z \cdot x \]

   if -2.8e9 < z < -2.99999999999999992e-290 or 3.99999999999999998e-90 < z < 4.20000000000000022e49

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 57.5

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.9

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

   7. Applied rewrites 27.9%

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

   if -2.99999999999999992e-290 < z < 3.99999999999999998e-90

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   3. 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 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 38.0%

      \[\leadsto t \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 36.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2800000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 2500000000000:\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2800000000.0) (* z x) (if (<= z 2500000000000.0) (* t y) (* z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2800000000.0) {
		tmp = z * x;
	} else if (z <= 2500000000000.0) {
		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 (z <= (-2800000000.0d0)) then
        tmp = z * x
    else if (z <= 2500000000000.0d0) 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 (z <= -2800000000.0) {
		tmp = z * x;
	} else if (z <= 2500000000000.0) {
		tmp = t * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2800000000.0:
		tmp = z * x
	elif z <= 2500000000000.0:
		tmp = t * y
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2800000000.0)
		tmp = Float64(z * x);
	elseif (z <= 2500000000000.0)
		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 (z <= -2800000000.0)
		tmp = z * x;
	elseif (z <= 2500000000000.0)
		tmp = t * y;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2800000000.0], N[(z * x), $MachinePrecision], If[LessEqual[z, 2500000000000.0], N[(t * y), $MachinePrecision], N[(z * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8e9 or 2.5e12 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 42.2%

      \[\leadsto z \cdot x \]

   if -2.8e9 < z < 2.5e12

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   3. 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 59.3

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

   4. Applied rewrites 59.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 35.2%

      \[\leadsto t \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 36.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-63}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9e-63) (* t y) (if (<= y 6.2e-76) x (* t y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-63) {
		tmp = t * y;
	} else if (y <= 6.2e-76) {
		tmp = x;
	} else {
		tmp = t * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-9d-63)) then
        tmp = t * y
    else if (y <= 6.2d-76) then
        tmp = x
    else
        tmp = t * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-63) {
		tmp = t * y;
	} else if (y <= 6.2e-76) {
		tmp = x;
	} else {
		tmp = t * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9e-63:
		tmp = t * y
	elif y <= 6.2e-76:
		tmp = x
	else:
		tmp = t * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9e-63)
		tmp = Float64(t * y);
	elseif (y <= 6.2e-76)
		tmp = x;
	else
		tmp = Float64(t * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9e-63)
		tmp = t * y;
	elseif (y <= 6.2e-76)
		tmp = x;
	else
		tmp = t * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9e-63], N[(t * y), $MachinePrecision], If[LessEqual[y, 6.2e-76], x, N[(t * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.9999999999999999e-63 or 6.19999999999999939e-76 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   3. 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 68.5

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 37.9%

      \[\leadsto t \cdot y \]

   if -8.9999999999999999e-63 < y < 6.19999999999999939e-76

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      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 94.0

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 35.3%

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

Alternative 13: 17.8% 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. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

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

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

   4. lower--.f64 N/A

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

   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 59.4

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

4. Applied rewrites 59.4%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 17.8%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025119 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64
  (+ x (* (- y z) (- t x))))