Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Percentage Accurate: 67.9% → 88.6%

Time: 12.3s

Alternatives: 21

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+204} \lor \neg \left(t \leq 4.6 \cdot 10^{+200}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -4.2e+204) (not (<= t 4.6e+200)))
   (- y (* (/ (- y x) t) (- z a)))
   (fma (/ (- z t) (- a t)) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.2e+204) || !(t <= 4.6e+200)) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = fma(((z - t) / (a - t)), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -4.2e+204) || !(t <= 4.6e+200))
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	else
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -4.2e+204], N[Not[LessEqual[t, 4.6e+200]], $MachinePrecision]], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.2000000000000001e204 or 4.60000000000000006e200 < t

   1. Initial program 22.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 53.0

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

   7. Applied rewrites 53.0%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 95.4

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

   10. Applied rewrites 95.4%

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

   if -4.2000000000000001e204 < t < 4.60000000000000006e200

   1. Initial program 84.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 91.0

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

   4. Applied rewrites 91.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+204} \lor \neg \left(t \leq 4.6 \cdot 10^{+200}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-14}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+66}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- y (* (/ (- y x) t) (- z a)))))
   (if (<= t -9.5e+121)
     t_1
     (if (<= t -8.5e-14)
       (+ x (/ (* (- z t) y) (- a t)))
       (if (<= t 5e+66) (+ x (/ (* (- y x) z) (- a t))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (((y - x) / t) * (z - a));
	double tmp;
	if (t <= -9.5e+121) {
		tmp = t_1;
	} else if (t <= -8.5e-14) {
		tmp = x + (((z - t) * y) / (a - t));
	} else if (t <= 5e+66) {
		tmp = x + (((y - x) * z) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
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), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y - (((y - x) / t) * (z - a))
    if (t <= (-9.5d+121)) then
        tmp = t_1
    else if (t <= (-8.5d-14)) then
        tmp = x + (((z - t) * y) / (a - t))
    else if (t <= 5d+66) then
        tmp = x + (((y - x) * z) / (a - t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y - (((y - x) / t) * (z - a));
	double tmp;
	if (t <= -9.5e+121) {
		tmp = t_1;
	} else if (t <= -8.5e-14) {
		tmp = x + (((z - t) * y) / (a - t));
	} else if (t <= 5e+66) {
		tmp = x + (((y - x) * z) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y - (((y - x) / t) * (z - a))
	tmp = 0
	if t <= -9.5e+121:
		tmp = t_1
	elif t <= -8.5e-14:
		tmp = x + (((z - t) * y) / (a - t))
	elif t <= 5e+66:
		tmp = x + (((y - x) * z) / (a - t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)))
	tmp = 0.0
	if (t <= -9.5e+121)
		tmp = t_1;
	elseif (t <= -8.5e-14)
		tmp = Float64(x + Float64(Float64(Float64(z - t) * y) / Float64(a - t)));
	elseif (t <= 5e+66)
		tmp = Float64(x + Float64(Float64(Float64(y - x) * z) / Float64(a - t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y - (((y - x) / t) * (z - a));
	tmp = 0.0;
	if (t <= -9.5e+121)
		tmp = t_1;
	elseif (t <= -8.5e-14)
		tmp = x + (((z - t) * y) / (a - t));
	elseif (t <= 5e+66)
		tmp = x + (((y - x) * z) / (a - t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.5e+121], t$95$1, If[LessEqual[t, -8.5e-14], N[(x + N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e+66], N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.49999999999999949e121 or 4.99999999999999991e66 < t

   1. Initial program 38.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 63.3

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 63.3

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

   7. Applied rewrites 63.3%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 81.9

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

   10. Applied rewrites 81.9%

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

   if -9.49999999999999949e121 < t < -8.50000000000000038e-14

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 77.4

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

   5. Applied rewrites 77.4%

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

   if -8.50000000000000038e-14 < t < 4.99999999999999991e66

   1. Initial program 90.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 82.8

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

   5. Applied rewrites 82.8%

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

Alternative 3: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+49} \lor \neg \left(t \leq 5 \cdot 10^{+66}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -4.1e+49) (not (<= t 5e+66)))
   (- y (* (/ (- y x) t) (- z a)))
   (+ x (/ (* (- y x) z) (- a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.1e+49) || !(t <= 5e+66)) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = x + (((y - x) * z) / (a - t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
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), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-4.1d+49)) .or. (.not. (t <= 5d+66))) then
        tmp = y - (((y - x) / t) * (z - a))
    else
        tmp = x + (((y - x) * z) / (a - t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.1e+49) || !(t <= 5e+66)) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = x + (((y - x) * z) / (a - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -4.1e+49) or not (t <= 5e+66):
		tmp = y - (((y - x) / t) * (z - a))
	else:
		tmp = x + (((y - x) * z) / (a - t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -4.1e+49) || !(t <= 5e+66))
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(y - x) * z) / Float64(a - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -4.1e+49) || ~((t <= 5e+66)))
		tmp = y - (((y - x) / t) * (z - a));
	else
		tmp = x + (((y - x) * z) / (a - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -4.1e+49], N[Not[LessEqual[t, 5e+66]], $MachinePrecision]], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.1e49 or 4.99999999999999991e66 < t

   1. Initial program 46.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 68.0

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

   4. Applied rewrites 68.0%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 67.9

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 80.0

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

   10. Applied rewrites 80.0%

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

   if -4.1e49 < t < 4.99999999999999991e66

   1. Initial program 89.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 80.1

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

   5. Applied rewrites 80.1%

      \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+49} \lor \neg \left(t \leq 5 \cdot 10^{+66}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-29} \lor \neg \left(a \leq 2.9 \cdot 10^{-51}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2e-29) (not (<= a 2.9e-51)))
   (fma (- z t) (/ (- y x) a) x)
   (- y (* (/ (- y x) t) (- z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2e-29) || !(a <= 2.9e-51)) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = y - (((y - x) / t) * (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2e-29) || !(a <= 2.9e-51))
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2e-29], N[Not[LessEqual[a, 2.9e-51]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999989e-29 or 2.89999999999999973e-51 < a

   1. Initial program 75.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -1.99999999999999989e-29 < a < 2.89999999999999973e-51

   1. Initial program 71.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 77.4

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

   7. Applied rewrites 77.4%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 78.7

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

   10. Applied rewrites 78.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-29} \lor \neg \left(a \leq 2.9 \cdot 10^{-51}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -3.6e+102)
     t_1
     (if (<= t -1.05e-254)
       (fma (/ y a) z x)
       (if (<= t 9.2e+142) (fma (- x) (/ z a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -3.6e+102) {
		tmp = t_1;
	} else if (t <= -1.05e-254) {
		tmp = fma((y / a), z, x);
	} else if (t <= 9.2e+142) {
		tmp = fma(-x, (z / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -3.6e+102)
		tmp = t_1;
	elseif (t <= -1.05e-254)
		tmp = fma(Float64(y / a), z, x);
	elseif (t <= 9.2e+142)
		tmp = fma(Float64(-x), Float64(z / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -3.6e+102], t$95$1, If[LessEqual[t, -1.05e-254], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t, 9.2e+142], N[((-x) * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.6000000000000002e102 or 9.20000000000000009e142 < t

   1. Initial program 42.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 66.7

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 38.2%

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

   if -3.6000000000000002e102 < t < -1.04999999999999998e-254

   1. Initial program 94.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.5%

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

   if -1.04999999999999998e-254 < t < 9.20000000000000009e142

   1. Initial program 81.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{a} \cdot \left(y - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2e-29)
   (fma (- z t) (/ (- y x) a) x)
   (if (<= a 4.8e-42)
     (- y (* (/ (- y x) t) (- z a)))
     (+ x (* (/ (- z t) a) (- y x))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2e-29) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else if (a <= 4.8e-42) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = x + (((z - t) / a) * (y - x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2e-29)
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	elseif (a <= 4.8e-42)
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(z - t) / a) * Float64(y - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2e-29], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 4.8e-42], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.99999999999999989e-29

   1. Initial program 80.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -1.99999999999999989e-29 < a < 4.80000000000000005e-42

   1. Initial program 71.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 77.4

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

   7. Applied rewrites 77.4%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 78.7

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

   10. Applied rewrites 78.7%

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

   if 4.80000000000000005e-42 < a

   1. Initial program 71.4%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{a} \cdot \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-42} \lor \neg \left(a \leq 1.15 \cdot 10^{-52}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.5e-42) (not (<= a 1.15e-52)))
   (fma (- z t) (/ (- y x) a) x)
   (* y (/ (- z t) (- a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.5e-42) || !(a <= 1.15e-52)) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = y * ((z - t) / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.5e-42) || !(a <= 1.15e-52))
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.5e-42], N[Not[LessEqual[a, 1.15e-52]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.50000000000000001e-42 or 1.14999999999999997e-52 < a

   1. Initial program 75.7%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -2.50000000000000001e-42 < a < 1.14999999999999997e-52

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 60.4

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 67.0%

      \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-42} \lor \neg \left(a \leq 1.15 \cdot 10^{-52}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{x}{a}, x\right)\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ x a) x)))
   (if (<= a -4.1e+15)
     t_1
     (if (<= a 3.7e-202) (* x (/ z t)) (if (<= a 3.2e-5) (/ (* y z) a) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, (x / a), x);
	double tmp;
	if (a <= -4.1e+15) {
		tmp = t_1;
	} else if (a <= 3.7e-202) {
		tmp = x * (z / t);
	} else if (a <= 3.2e-5) {
		tmp = (y * z) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(x / a), x)
	tmp = 0.0
	if (a <= -4.1e+15)
		tmp = t_1;
	elseif (a <= 3.7e-202)
		tmp = Float64(x * Float64(z / t));
	elseif (a <= 3.2e-5)
		tmp = Float64(Float64(y * z) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(x / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -4.1e+15], t$95$1, If[LessEqual[a, 3.7e-202], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.2e-5], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.1e15 or 3.19999999999999986e-5 < a

   1. Initial program 73.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 44.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 44.5%

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

   if -4.1e15 < a < 3.69999999999999991e-202

   1. Initial program 72.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 27.5

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

   5. Applied rewrites 27.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 33.3%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 29.1%

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

   if 3.69999999999999991e-202 < a < 3.19999999999999986e-5

   1. Initial program 81.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 43.6

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.0%

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+45} \lor \neg \left(t \leq 8.8 \cdot 10^{+30}\right):\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -5.5e+45) (not (<= t 8.8e+30)))
   (* y (/ (- z t) (- a t)))
   (fma (/ z a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.5e+45) || !(t <= 8.8e+30)) {
		tmp = y * ((z - t) / (a - t));
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -5.5e+45) || !(t <= 8.8e+30))
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	else
		tmp = fma(Float64(z / a), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -5.5e+45], N[Not[LessEqual[t, 8.8e+30]], $MachinePrecision]], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.5000000000000001e45 or 8.7999999999999999e30 < t

   1. Initial program 50.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 47.3

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

   7. Applied rewrites 47.3%

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

   9. Applied rewrites 65.1%

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

   if -5.5000000000000001e45 < t < 8.7999999999999999e30

   1. Initial program 90.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 76.3

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

   7. Applied rewrites 76.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+45} \lor \neg \left(t \leq 8.8 \cdot 10^{+30}\right):\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -0.23:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -0.23)
     t_1
     (if (<= t 2.9e-126)
       (* y (/ z a))
       (if (<= t 1.65e+176) (* x (/ z t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -0.23) {
		tmp = t_1;
	} else if (t <= 2.9e-126) {
		tmp = y * (z / a);
	} else if (t <= 1.65e+176) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -0.23)
		tmp = t_1;
	elseif (t <= 2.9e-126)
		tmp = Float64(y * Float64(z / a));
	elseif (t <= 1.65e+176)
		tmp = Float64(x * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -0.23], t$95$1, If[LessEqual[t, 2.9e-126], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.65e+176], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.23000000000000001 or 1.65000000000000012e176 < t

   1. Initial program 51.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.9

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

   4. Applied rewrites 71.9%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 36.3%

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

   if -0.23000000000000001 < t < 2.89999999999999988e-126

   1. Initial program 93.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 42.9

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

   7. Applied rewrites 42.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 35.2%

       \[\leadsto y \cdot \color{blue}{\frac{z}{a}} \]

   if 2.89999999999999988e-126 < t < 1.65000000000000012e176

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 46.2

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 30.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 24.3%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.23:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -0.23:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -0.23)
     t_1
     (if (<= t 2.9e-126)
       (/ (* y z) a)
       (if (<= t 1.65e+176) (* x (/ z t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -0.23) {
		tmp = t_1;
	} else if (t <= 2.9e-126) {
		tmp = (y * z) / a;
	} else if (t <= 1.65e+176) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -0.23)
		tmp = t_1;
	elseif (t <= 2.9e-126)
		tmp = Float64(Float64(y * z) / a);
	elseif (t <= 1.65e+176)
		tmp = Float64(x * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -0.23], t$95$1, If[LessEqual[t, 2.9e-126], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 1.65e+176], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.23000000000000001 or 1.65000000000000012e176 < t

   1. Initial program 51.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.9

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

   4. Applied rewrites 71.9%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 36.3%

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

   if -0.23000000000000001 < t < 2.89999999999999988e-126

   1. Initial program 93.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.3%

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]

   if 2.89999999999999988e-126 < t < 1.65000000000000012e176

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 46.2

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 30.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 24.3%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.23:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+49} \lor \neg \left(t \leq 5.2 \cdot 10^{+66}\right):\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -4.3e+49) (not (<= t 5.2e+66)))
   (* (- y) (/ (- z t) t))
   (fma (/ z a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.3e+49) || !(t <= 5.2e+66)) {
		tmp = -y * ((z - t) / t);
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -4.3e+49) || !(t <= 5.2e+66))
		tmp = Float64(Float64(-y) * Float64(Float64(z - t) / t));
	else
		tmp = fma(Float64(z / a), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -4.3e+49], N[Not[LessEqual[t, 5.2e+66]], $MachinePrecision]], N[((-y) * N[(N[(z - t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.2999999999999999e49 or 5.20000000000000024e66 < t

   1. Initial program 46.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 68.0

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

   4. Applied rewrites 68.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 45.2

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

   7. Applied rewrites 45.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 55.8%

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

   if -4.2999999999999999e49 < t < 5.20000000000000024e66

   1. Initial program 89.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+49} \lor \neg \left(t \leq 5.2 \cdot 10^{+66}\right):\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+49} \lor \neg \left(t \leq 3.1 \cdot 10^{+141}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.9e+49) (not (<= t 3.1e+141)))
   (* (- t) (/ y (- a t)))
   (fma (/ z a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.9e+49) || !(t <= 3.1e+141)) {
		tmp = -t * (y / (a - t));
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.9e+49) || !(t <= 3.1e+141))
		tmp = Float64(Float64(-t) * Float64(y / Float64(a - t)));
	else
		tmp = fma(Float64(z / a), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.9e+49], N[Not[LessEqual[t, 3.1e+141]], $MachinePrecision]], N[((-t) * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9e49 or 3.10000000000000004e141 < t

   1. Initial program 47.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 46.7

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 43.4%

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

   if -2.9e49 < t < 3.10000000000000004e141

   1. Initial program 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 71.3

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

   7. Applied rewrites 71.3%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+49} \lor \neg \left(t \leq 3.1 \cdot 10^{+141}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-175} \lor \neg \left(a \leq 6.1 \cdot 10^{-93}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y - x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.2e-175) (not (<= a 6.1e-93)))
   (fma (/ z a) (- y x) x)
   (* (- z) (/ (- y x) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.2e-175) || !(a <= 6.1e-93)) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = -z * ((y - x) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.2e-175) || !(a <= 6.1e-93))
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = Float64(Float64(-z) * Float64(Float64(y - x) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.2e-175], N[Not[LessEqual[a, 6.1e-93]], $MachinePrecision]], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[((-z) * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.2e-175 or 6.09999999999999971e-93 < a

   1. Initial program 77.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 65.9

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

   7. Applied rewrites 65.9%

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

   if -4.2e-175 < a < 6.09999999999999971e-93

   1. Initial program 66.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 58.1

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

   7. Applied rewrites 58.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 53.9%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-175} \lor \neg \left(a \leq 6.1 \cdot 10^{-93}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y - x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -5.5e+45)
   (* (- z t) (/ y (- a t)))
   (if (<= t 5.2e+66) (fma (/ z a) (- y x) x) (* (- y) (/ (- z t) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -5.5e+45) {
		tmp = (z - t) * (y / (a - t));
	} else if (t <= 5.2e+66) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = -y * ((z - t) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -5.5e+45)
		tmp = Float64(Float64(z - t) * Float64(y / Float64(a - t)));
	elseif (t <= 5.2e+66)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = Float64(Float64(-y) * Float64(Float64(z - t) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -5.5e+45], N[(N[(z - t), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.2e+66], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[((-y) * N[(N[(z - t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.5000000000000001e45

   1. Initial program 57.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 63.1

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

   5. Applied rewrites 63.1%

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

   if -5.5000000000000001e45 < t < 5.20000000000000024e66

   1. Initial program 89.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 74.3

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

   7. Applied rewrites 74.3%

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

   if 5.20000000000000024e66 < t

   1. Initial program 34.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 59.8

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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 39.3

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 49.8%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.05 \cdot 10^{+134} \lor \neg \left(t \leq 9.5 \cdot 10^{+142}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3.05e+134) (not (<= t 9.5e+142)))
   (fma 1.0 (- y x) x)
   (fma (/ z a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.05e+134) || !(t <= 9.5e+142)) {
		tmp = fma(1.0, (y - x), x);
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3.05e+134) || !(t <= 9.5e+142))
		tmp = fma(1.0, Float64(y - x), x);
	else
		tmp = fma(Float64(z / a), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3.05e+134], N[Not[LessEqual[t, 9.5e+142]], $MachinePrecision]], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.04999999999999989e134 or 9.50000000000000001e142 < t

   1. Initial program 36.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 40.0%

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

   if -3.04999999999999989e134 < t < 9.50000000000000001e142

   1. Initial program 85.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 66.5

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

   7. Applied rewrites 66.5%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.05 \cdot 10^{+134} \lor \neg \left(t \leq 9.5 \cdot 10^{+142}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.05 \cdot 10^{+134} \lor \neg \left(t \leq 9.5 \cdot 10^{+142}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3.05e+134) (not (<= t 9.5e+142)))
   (fma 1.0 (- y x) x)
   (fma (/ (- y x) a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.05e+134) || !(t <= 9.5e+142)) {
		tmp = fma(1.0, (y - x), x);
	} else {
		tmp = fma(((y - x) / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3.05e+134) || !(t <= 9.5e+142))
		tmp = fma(1.0, Float64(y - x), x);
	else
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3.05e+134], N[Not[LessEqual[t, 9.5e+142]], $MachinePrecision]], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.04999999999999989e134 or 9.50000000000000001e142 < t

   1. Initial program 36.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 40.0%

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

   if -3.04999999999999989e134 < t < 9.50000000000000001e142

   1. Initial program 85.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 65.2

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

   5. Applied rewrites 65.2%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.05 \cdot 10^{+134} \lor \neg \left(t \leq 9.5 \cdot 10^{+142}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+102} \lor \neg \left(t \leq 9.5 \cdot 10^{+142}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3.6e+102) (not (<= t 9.5e+142)))
   (fma 1.0 (- y x) x)
   (fma (/ y a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.6e+102) || !(t <= 9.5e+142)) {
		tmp = fma(1.0, (y - x), x);
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3.6e+102) || !(t <= 9.5e+142))
		tmp = fma(1.0, Float64(y - x), x);
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3.6e+102], N[Not[LessEqual[t, 9.5e+142]], $MachinePrecision]], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.6000000000000002e102 or 9.50000000000000001e142 < t

   1. Initial program 42.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 66.7

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 38.2%

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

   if -3.6000000000000002e102 < t < 9.50000000000000001e142

   1. Initial program 86.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.5%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+102} \lor \neg \left(t \leq 9.5 \cdot 10^{+142}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-12} \lor \neg \left(z \leq 2.5 \cdot 10^{-65}\right):\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.8e-12) (not (<= z 2.5e-65)))
   (* x (/ z t))
   (fma 1.0 (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.8e-12) || !(z <= 2.5e-65)) {
		tmp = x * (z / t);
	} else {
		tmp = fma(1.0, (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.8e-12) || !(z <= 2.5e-65))
		tmp = Float64(x * Float64(z / t));
	else
		tmp = fma(1.0, Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.8e-12], N[Not[LessEqual[z, 2.5e-65]], $MachinePrecision]], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.79999999999999996e-12 or 2.49999999999999991e-65 < z

   1. Initial program 75.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 45.2

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 27.1%

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

   if -3.79999999999999996e-12 < z < 2.49999999999999991e-65

   1. Initial program 72.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 80.0

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 27.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-12} \lor \neg \left(z \leq 2.5 \cdot 10^{-65}\right):\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 19.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, y - x, x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma 1.0 (- y x) x))

C

double code(double x, double y, double z, double t, double a) {
	return fma(1.0, (y - x), x);
}

Julia

function code(x, y, z, t, a)
	return fma(1.0, Float64(y - x), x)
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 73.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 84.7

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

4. Applied rewrites 84.7%

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

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
6. Step-by-step derivation

7. Applied rewrites 16.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, y - x, x\right) \]
8. Add Preprocessing

Alternative 21: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

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, a)
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), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. mul-1-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

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

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

   10. mul-1-neg N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 39.9

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

5. Applied rewrites 39.9%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 2.9%

   \[\leadsto 0 \cdot \color{blue}{x} \]

9. Taylor expanded in x around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 2.9%

    \[\leadsto 0 \]

12. Final simplification 2.9%

    \[\leadsto 0 \]
13. Add Preprocessing

Developer Target 1: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
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), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024364 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

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