Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.1% → 95.0%

Time: 8.7s

Alternatives: 18

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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 - z) * ((t - x) / (a - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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 - z) * ((t - x) / (a - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-288} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (or (<= t_1 -1e-288) (not (<= t_1 0.0)))
     (fma (- t x) (/ (- y z) (- a z)) x)
     (- t (* (/ (- t x) z) (- y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if ((t_1 <= -1e-288) || !(t_1 <= 0.0)) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else {
		tmp = t - (((t - x) / z) * (y - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if ((t_1 <= -1e-288) || !(t_1 <= 0.0))
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	else
		tmp = Float64(t - Float64(Float64(Float64(t - x) / z) * Float64(y - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-288], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t - N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000006e-288 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 88.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 94.5

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

   4. Applied rewrites 94.5%

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

   if -1.00000000000000006e-288 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 6.8

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

   4. Applied rewrites 6.8%

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

   5. Taylor expanded in z around inf

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

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

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. associate-/l* N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-288} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(t - x\right)\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -41000000:\\ \;\;\;\;\frac{y - a}{z} \cdot x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+165}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- t x))))
   (if (<= z -7.2e+157)
     t_1
     (if (<= z -41000000.0)
       (* (/ (- y a) z) x)
       (if (<= z 6.2e-30)
         (+ x (/ (* y t) a))
         (if (<= z 1.12e+165) (* (/ (- x t) z) y) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (t - x);
	double tmp;
	if (z <= -7.2e+157) {
		tmp = t_1;
	} else if (z <= -41000000.0) {
		tmp = ((y - a) / z) * x;
	} else if (z <= 6.2e-30) {
		tmp = x + ((y * t) / a);
	} else if (z <= 1.12e+165) {
		tmp = ((x - t) / z) * y;
	} 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 + (t - x)
    if (z <= (-7.2d+157)) then
        tmp = t_1
    else if (z <= (-41000000.0d0)) then
        tmp = ((y - a) / z) * x
    else if (z <= 6.2d-30) then
        tmp = x + ((y * t) / a)
    else if (z <= 1.12d+165) then
        tmp = ((x - t) / z) * y
    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 + (t - x);
	double tmp;
	if (z <= -7.2e+157) {
		tmp = t_1;
	} else if (z <= -41000000.0) {
		tmp = ((y - a) / z) * x;
	} else if (z <= 6.2e-30) {
		tmp = x + ((y * t) / a);
	} else if (z <= 1.12e+165) {
		tmp = ((x - t) / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (t - x)
	tmp = 0
	if z <= -7.2e+157:
		tmp = t_1
	elif z <= -41000000.0:
		tmp = ((y - a) / z) * x
	elif z <= 6.2e-30:
		tmp = x + ((y * t) / a)
	elif z <= 1.12e+165:
		tmp = ((x - t) / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(t - x))
	tmp = 0.0
	if (z <= -7.2e+157)
		tmp = t_1;
	elseif (z <= -41000000.0)
		tmp = Float64(Float64(Float64(y - a) / z) * x);
	elseif (z <= 6.2e-30)
		tmp = Float64(x + Float64(Float64(y * t) / a));
	elseif (z <= 1.12e+165)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (t - x);
	tmp = 0.0;
	if (z <= -7.2e+157)
		tmp = t_1;
	elseif (z <= -41000000.0)
		tmp = ((y - a) / z) * x;
	elseif (z <= 6.2e-30)
		tmp = x + ((y * t) / a);
	elseif (z <= 1.12e+165)
		tmp = ((x - t) / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+157], t$95$1, If[LessEqual[z, -41000000.0], N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 6.2e-30], N[(x + N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.12e+165], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.20000000000000049e157 or 1.1200000000000001e165 < z

   1. Initial program 55.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 58.1

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

   5. Applied rewrites 58.1%

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

   if -7.20000000000000049e157 < z < -4.1e7

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 38.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.8%

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

   if -4.1e7 < z < 6.19999999999999982e-30

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.6%

      \[\leadsto x + \frac{y \cdot t}{a} \]

   if 6.19999999999999982e-30 < z < 1.1200000000000001e165

   1. Initial program 86.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 47.3%

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

Alternative 3: 47.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(t - x\right)\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -41000000:\\ \;\;\;\;\frac{y - a}{z} \cdot x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+165}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- t x))))
   (if (<= z -7.2e+157)
     t_1
     (if (<= z -41000000.0)
       (* (/ (- y a) z) x)
       (if (<= z 6.2e-30)
         (fma y (/ t a) x)
         (if (<= z 1.12e+165) (* (/ (- x t) z) y) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (t - x);
	double tmp;
	if (z <= -7.2e+157) {
		tmp = t_1;
	} else if (z <= -41000000.0) {
		tmp = ((y - a) / z) * x;
	} else if (z <= 6.2e-30) {
		tmp = fma(y, (t / a), x);
	} else if (z <= 1.12e+165) {
		tmp = ((x - t) / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(t - x))
	tmp = 0.0
	if (z <= -7.2e+157)
		tmp = t_1;
	elseif (z <= -41000000.0)
		tmp = Float64(Float64(Float64(y - a) / z) * x);
	elseif (z <= 6.2e-30)
		tmp = fma(y, Float64(t / a), x);
	elseif (z <= 1.12e+165)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+157], t$95$1, If[LessEqual[z, -41000000.0], N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 6.2e-30], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.12e+165], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.20000000000000049e157 or 1.1200000000000001e165 < z

   1. Initial program 55.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 58.1

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

   5. Applied rewrites 58.1%

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

   if -7.20000000000000049e157 < z < -4.1e7

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 38.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.8%

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

   if -4.1e7 < z < 6.19999999999999982e-30

   1. Initial program 90.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.9%

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

   if 6.19999999999999982e-30 < z < 1.1200000000000001e165

   1. Initial program 86.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 47.3%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+157}:\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{elif}\;z \leq -41000000:\\ \;\;\;\;\frac{y - a}{z} \cdot x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+165}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a - y}{z}, t, t\right)\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -41000000:\\ \;\;\;\;\frac{y - a}{z} \cdot x\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- a y) z) t t)))
   (if (<= z -2.6e+157)
     t_1
     (if (<= z -41000000.0)
       (* (/ (- y a) z) x)
       (if (<= z 4.6e-30) (+ x (/ (* y t) a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((a - y) / z), t, t);
	double tmp;
	if (z <= -2.6e+157) {
		tmp = t_1;
	} else if (z <= -41000000.0) {
		tmp = ((y - a) / z) * x;
	} else if (z <= 4.6e-30) {
		tmp = x + ((y * t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(a - y) / z), t, t)
	tmp = 0.0
	if (z <= -2.6e+157)
		tmp = t_1;
	elseif (z <= -41000000.0)
		tmp = Float64(Float64(Float64(y - a) / z) * x);
	elseif (z <= 4.6e-30)
		tmp = Float64(x + Float64(Float64(y * t) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision] * t + t), $MachinePrecision]}, If[LessEqual[z, -2.6e+157], t$95$1, If[LessEqual[z, -41000000.0], N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 4.6e-30], N[(x + N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.60000000000000011e157 or 4.59999999999999968e-30 < z

   1. Initial program 67.1%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 27.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.6%

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

   if -2.60000000000000011e157 < z < -4.1e7

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 38.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.8%

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

   if -4.1e7 < z < 4.59999999999999968e-30

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.6%

      \[\leadsto x + \frac{y \cdot t}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -15200 \lor \neg \left(z \leq 4.6 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -15200.0) (not (<= z 4.6e-30)))
   (fma (- (- t x)) (/ (- y a) z) t)
   (fma (- t x) (/ (- y z) a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -15200.0) || !(z <= 4.6e-30)) {
		tmp = fma(-(t - x), ((y - a) / z), t);
	} else {
		tmp = fma((t - x), ((y - z) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -15200.0) || !(z <= 4.6e-30))
		tmp = fma(Float64(-Float64(t - x)), Float64(Float64(y - a) / z), t);
	else
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -15200.0], N[Not[LessEqual[z, 4.6e-30]], $MachinePrecision]], N[((-N[(t - x), $MachinePrecision]) * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -15200 or 4.59999999999999968e-30 < z

   1. Initial program 67.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 81.2

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

   5. Applied rewrites 81.2%

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

   if -15200 < z < 4.59999999999999968e-30

   1. Initial program 90.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 79.7

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

   7. Applied rewrites 79.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -15200 \lor \neg \left(z \leq 4.6 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(t - x\right)\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+165}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- t x))))
   (if (<= z -1.25e+165)
     t_1
     (if (<= z 6.2e-30)
       (fma y (/ t a) x)
       (if (<= z 1.12e+165) (* (/ (- x t) z) y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (t - x);
	double tmp;
	if (z <= -1.25e+165) {
		tmp = t_1;
	} else if (z <= 6.2e-30) {
		tmp = fma(y, (t / a), x);
	} else if (z <= 1.12e+165) {
		tmp = ((x - t) / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(t - x))
	tmp = 0.0
	if (z <= -1.25e+165)
		tmp = t_1;
	elseif (z <= 6.2e-30)
		tmp = fma(y, Float64(t / a), x);
	elseif (z <= 1.12e+165)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.25e+165], t$95$1, If[LessEqual[z, 6.2e-30], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.12e+165], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.24999999999999993e165 or 1.1200000000000001e165 < z

   1. Initial program 54.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -1.24999999999999993e165 < z < 6.19999999999999982e-30

   1. Initial program 86.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 67.5

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 58.6%

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

   if 6.19999999999999982e-30 < z < 1.1200000000000001e165

   1. Initial program 86.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 47.3%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+165}:\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+165}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -15200:\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -15200.0)
   (- t (* (/ (- t x) z) (- y a)))
   (if (<= z 4.6e-30)
     (fma (- t x) (/ (- y z) a) x)
     (fma (- (- t x)) (/ (- y a) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -15200.0) {
		tmp = t - (((t - x) / z) * (y - a));
	} else if (z <= 4.6e-30) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else {
		tmp = fma(-(t - x), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -15200.0)
		tmp = Float64(t - Float64(Float64(Float64(t - x) / z) * Float64(y - a)));
	elseif (z <= 4.6e-30)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	else
		tmp = fma(Float64(-Float64(t - x)), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -15200.0], N[(t - N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.6e-30], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[((-N[(t - x), $MachinePrecision]) * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -15200

   1. Initial program 61.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 66.3

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

   4. Applied rewrites 66.3%

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

   5. Taylor expanded in z around inf

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

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

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. associate-/l* N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 82.7

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

   7. Applied rewrites 82.7%

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

   if -15200 < z < 4.59999999999999968e-30

   1. Initial program 90.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 79.7

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

   7. Applied rewrites 79.7%

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

   if 4.59999999999999968e-30 < z

   1. Initial program 74.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 79.9

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

   5. Applied rewrites 79.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -15200:\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.7e-40)
   (fma (/ (- x t) z) y t)
   (if (<= z 6.2e-30)
     (fma (- t x) (/ (- y z) a) x)
     (fma (- (- t x)) (/ y z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.7e-40) {
		tmp = fma(((x - t) / z), y, t);
	} else if (z <= 6.2e-30) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else {
		tmp = fma(-(t - x), (y / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.7e-40)
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	elseif (z <= 6.2e-30)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	else
		tmp = fma(Float64(-Float64(t - x)), Float64(y / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.7e-40], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], If[LessEqual[z, 6.2e-30], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[((-N[(t - x), $MachinePrecision]) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.69999999999999998e-40

   1. Initial program 65.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.0%

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

   if -3.69999999999999998e-40 < z < 6.19999999999999982e-30

   1. Initial program 90.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if 6.19999999999999982e-30 < z

   1. Initial program 74.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 74.8%

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

Alternative 9: 28.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(t - x\right)\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;z \leq 910000:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- t x))))
   (if (<= z -2.5e+157)
     t_1
     (if (<= z -2.3e-44)
       (* (/ x z) y)
       (if (<= z 910000.0) (/ (* t y) a) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (t - x);
	double tmp;
	if (z <= -2.5e+157) {
		tmp = t_1;
	} else if (z <= -2.3e-44) {
		tmp = (x / z) * y;
	} else if (z <= 910000.0) {
		tmp = (t * y) / a;
	} 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 + (t - x)
    if (z <= (-2.5d+157)) then
        tmp = t_1
    else if (z <= (-2.3d-44)) then
        tmp = (x / z) * y
    else if (z <= 910000.0d0) then
        tmp = (t * y) / a
    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 + (t - x);
	double tmp;
	if (z <= -2.5e+157) {
		tmp = t_1;
	} else if (z <= -2.3e-44) {
		tmp = (x / z) * y;
	} else if (z <= 910000.0) {
		tmp = (t * y) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (t - x)
	tmp = 0
	if z <= -2.5e+157:
		tmp = t_1
	elif z <= -2.3e-44:
		tmp = (x / z) * y
	elif z <= 910000.0:
		tmp = (t * y) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(t - x))
	tmp = 0.0
	if (z <= -2.5e+157)
		tmp = t_1;
	elseif (z <= -2.3e-44)
		tmp = Float64(Float64(x / z) * y);
	elseif (z <= 910000.0)
		tmp = Float64(Float64(t * y) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (t - x);
	tmp = 0.0;
	if (z <= -2.5e+157)
		tmp = t_1;
	elseif (z <= -2.3e-44)
		tmp = (x / z) * y;
	elseif (z <= 910000.0)
		tmp = (t * y) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.5e+157], t$95$1, If[LessEqual[z, -2.3e-44], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 910000.0], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.49999999999999988e157 or 9.1e5 < z

   1. Initial program 63.9%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 47.1

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

   5. Applied rewrites 47.1%

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

   if -2.49999999999999988e157 < z < -2.29999999999999998e-44

   1. Initial program 77.2%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 41.0%

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

   if -2.29999999999999998e-44 < z < 9.1e5

   1. Initial program 90.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 72.8

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

   7. Applied rewrites 72.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.4%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+157}:\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;z \leq 910000:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6600000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(t - x\right), \frac{y}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6600000.0)
   (fma (/ (- x t) z) y t)
   (if (<= z 6.2e-30) (fma (- t x) (/ y a) x) (fma (- (- t x)) (/ y z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6600000.0) {
		tmp = fma(((x - t) / z), y, t);
	} else if (z <= 6.2e-30) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = fma(-(t - x), (y / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6600000.0)
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	elseif (z <= 6.2e-30)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = fma(Float64(-Float64(t - x)), Float64(y / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6600000.0], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], If[LessEqual[z, 6.2e-30], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[((-N[(t - x), $MachinePrecision]) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.6e6

   1. Initial program 61.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.9%

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

   if -6.6e6 < z < 6.19999999999999982e-30

   1. Initial program 90.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 6.19999999999999982e-30 < z

   1. Initial program 74.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 74.8%

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

Alternative 11: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6600000 \lor \neg \left(z \leq 6.2 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6600000.0) (not (<= z 6.2e-30)))
   (fma (/ (- x t) z) y t)
   (fma (- t x) (/ y a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6600000.0) || !(z <= 6.2e-30)) {
		tmp = fma(((x - t) / z), y, t);
	} else {
		tmp = fma((t - x), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6600000.0) || !(z <= 6.2e-30))
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	else
		tmp = fma(Float64(t - x), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6600000.0], N[Not[LessEqual[z, 6.2e-30]], $MachinePrecision]], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6e6 or 6.19999999999999982e-30 < z

   1. Initial program 67.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.1%

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

   if -6.6e6 < z < 6.19999999999999982e-30

   1. Initial program 90.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6600000 \lor \neg \left(z \leq 6.2 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1350 \lor \neg \left(z \leq 6.2 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1350.0) (not (<= z 6.2e-30)))
   (fma (/ (- x t) z) y t)
   (fma (/ (- t x) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1350.0) || !(z <= 6.2e-30)) {
		tmp = fma(((x - t) / z), y, t);
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1350.0) || !(z <= 6.2e-30))
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1350.0], N[Not[LessEqual[z, 6.2e-30]], $MachinePrecision]], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1350 or 6.19999999999999982e-30 < z

   1. Initial program 67.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.1%

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

   if -1350 < z < 6.19999999999999982e-30

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.0

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

   5. Applied rewrites 76.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1350 \lor \neg \left(z \leq 6.2 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+14} \lor \neg \left(a \leq 1.26 \cdot 10^{+150}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7e+14) (not (<= a 1.26e+150)))
   (fma y (/ t a) x)
   (fma (/ (- x t) z) y t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7e+14) || !(a <= 1.26e+150)) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7e+14) || !(a <= 1.26e+150))
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7e+14], N[Not[LessEqual[a, 1.26e+150]], $MachinePrecision]], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7e14 or 1.26e150 < a

   1. Initial program 87.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 79.4

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.3%

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

   if -7e14 < a < 1.26e150

   1. Initial program 74.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+14} \lor \neg \left(a \leq 1.26 \cdot 10^{+150}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+165} \lor \neg \left(z \leq 1.02 \cdot 10^{+165}\right):\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.25e+165) (not (<= z 1.02e+165)))
   (+ x (- t x))
   (fma y (/ t a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.25e+165) || !(z <= 1.02e+165)) {
		tmp = x + (t - x);
	} else {
		tmp = fma(y, (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.25e+165) || !(z <= 1.02e+165))
		tmp = Float64(x + Float64(t - x));
	else
		tmp = fma(y, Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.25e+165], N[Not[LessEqual[z, 1.02e+165]], $MachinePrecision]], N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.24999999999999993e165 or 1.02000000000000003e165 < z

   1. Initial program 54.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -1.24999999999999993e165 < z < 1.02000000000000003e165

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 90.1

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 60.8

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 53.2%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+165} \lor \neg \left(z \leq 1.02 \cdot 10^{+165}\right):\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-15} \lor \neg \left(x \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= x -8.5e-15) (not (<= x 2e-10))) (* (/ y z) x) (+ x (- t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x <= -8.5e-15) || !(x <= 2e-10)) {
		tmp = (y / z) * x;
	} else {
		tmp = x + (t - x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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 ((x <= (-8.5d-15)) .or. (.not. (x <= 2d-10))) then
        tmp = (y / z) * x
    else
        tmp = x + (t - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x <= -8.5e-15) || !(x <= 2e-10)) {
		tmp = (y / z) * x;
	} else {
		tmp = x + (t - x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x <= -8.5e-15) or not (x <= 2e-10):
		tmp = (y / z) * x
	else:
		tmp = x + (t - x)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((x <= -8.5e-15) || !(x <= 2e-10))
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(x + Float64(t - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x <= -8.5e-15) || ~((x <= 2e-10)))
		tmp = (y / z) * x;
	else
		tmp = x + (t - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[x, -8.5e-15], N[Not[LessEqual[x, 2e-10]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000007e-15 or 2.00000000000000007e-10 < x

   1. Initial program 75.2%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 32.8%

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

   if -8.50000000000000007e-15 < x < 2.00000000000000007e-10

   1. Initial program 82.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 34.4

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

   5. Applied rewrites 34.4%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-15} \lor \neg \left(x \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-10}:\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -8.5e-15)
   (* (/ x z) y)
   (if (<= x 2e-10) (+ x (- t x)) (* (/ y z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.5e-15) {
		tmp = (x / z) * y;
	} else if (x <= 2e-10) {
		tmp = x + (t - x);
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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 (x <= (-8.5d-15)) then
        tmp = (x / z) * y
    else if (x <= 2d-10) then
        tmp = x + (t - x)
    else
        tmp = (y / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.5e-15) {
		tmp = (x / z) * y;
	} else if (x <= 2e-10) {
		tmp = x + (t - x);
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -8.5e-15:
		tmp = (x / z) * y
	elif x <= 2e-10:
		tmp = x + (t - x)
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -8.5e-15)
		tmp = Float64(Float64(x / z) * y);
	elseif (x <= 2e-10)
		tmp = Float64(x + Float64(t - x));
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -8.5e-15)
		tmp = (x / z) * y;
	elseif (x <= 2e-10)
		tmp = x + (t - x);
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -8.5e-15], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 2e-10], N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.50000000000000007e-15

   1. Initial program 78.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 54.2

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 32.3%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 28.7%

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

   if -8.50000000000000007e-15 < x < 2.00000000000000007e-10

   1. Initial program 82.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 34.4

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

   5. Applied rewrites 34.4%

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

   if 2.00000000000000007e-10 < x

   1. Initial program 72.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 36.5%

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

Alternative 17: 19.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (t - x);
}

Python

def code(x, y, z, t, a):
	return x + (t - x)

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(t - x))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (t - x);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
4. Step-by-step derivation

   1. lower--.f64 22.3

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

5. Applied rewrites 22.3%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
6. Add Preprocessing

Alternative 18: 2.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ x + \left(-x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (- x)))

C

double code(double x, double y, double z, double t, double a) {
	return x + -x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, 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 + -x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + -x;
}

Python

def code(x, y, z, t, a):
	return x + -x

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(-x))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + -x;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + (-x)), $MachinePrecision]

Derivation

1. Initial program 79.1%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
4. Step-by-step derivation

   1. lower--.f64 22.3

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

5. Applied rewrites 22.3%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 2.8%

   \[\leadsto x + \left(-x\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024364 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))