Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 84.7% → 98.2%

Time: 10.9s

Alternatives: 12

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y - z\right)}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x\_m \cdot \left(\frac{y - z}{z + t} \cdot \frac{z + t}{t - z}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m (- y z)) (- t z))))
   (*
    x_s
    (if (<= t_1 0.0)
      (* x_m (* (/ (- y z) (+ z t)) (/ (+ z t) (- t z))))
      (if (<= t_1 5e+303) t_1 (* (/ x_m (- t z)) (- y z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x_m * (((y - z) / (z + t)) * ((z + t) / (t - z)));
	} else if (t_1 <= 5e+303) {
		tmp = t_1;
	} else {
		tmp = (x_m / (t - z)) * (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (y - z)) / (t - z)
    if (t_1 <= 0.0d0) then
        tmp = x_m * (((y - z) / (z + t)) * ((z + t) / (t - z)))
    else if (t_1 <= 5d+303) then
        tmp = t_1
    else
        tmp = (x_m / (t - z)) * (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x_m * (((y - z) / (z + t)) * ((z + t) / (t - z)));
	} else if (t_1 <= 5e+303) {
		tmp = t_1;
	} else {
		tmp = (x_m / (t - z)) * (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= 0.0:
		tmp = x_m * (((y - z) / (z + t)) * ((z + t) / (t - z)))
	elif t_1 <= 5e+303:
		tmp = t_1
	else:
		tmp = (x_m / (t - z)) * (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x_m * Float64(Float64(Float64(y - z) / Float64(z + t)) * Float64(Float64(z + t) / Float64(t - z))));
	elseif (t_1 <= 5e+303)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = x_m * (((y - z) / (z + t)) * ((z + t) / (t - z)));
	elseif (t_1 <= 5e+303)
		tmp = t_1;
	else
		tmp = (x_m / (t - z)) * (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 0.0], N[(x$95$m * N[(N[(N[(y - z), $MachinePrecision] / N[(z + t), $MachinePrecision]), $MachinePrecision] * N[(N[(z + t), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+303], t$95$1, N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -0.0

   1. Initial program 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. difference-of-squares-rev N/A

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

      9. lift-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-/r/ N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. difference-of-squares-rev N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. *-commutative N/A

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

      20. lift--.f64 N/A

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

      21. associate-/l* N/A

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

      22. associate-*r* N/A

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

   6. Applied rewrites 96.6%

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

   if -0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 4.9999999999999997e303

   1. Initial program 98.9%

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

   if 4.9999999999999997e303 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 36.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y - z\right)}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+289} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m (- y z)) (- t z))))
   (*
    x_s
    (if (or (<= t_1 -4e+289) (not (<= t_1 5e+303)))
      (* (/ x_m (- t z)) (- y z))
      t_1))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if ((t_1 <= -4e+289) || !(t_1 <= 5e+303)) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (y - z)) / (t - z)
    if ((t_1 <= (-4d+289)) .or. (.not. (t_1 <= 5d+303))) then
        tmp = (x_m / (t - z)) * (y - z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * (y - z)) / (t - z);
	double tmp;
	if ((t_1 <= -4e+289) || !(t_1 <= 5e+303)) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m * (y - z)) / (t - z)
	tmp = 0
	if (t_1 <= -4e+289) or not (t_1 <= 5e+303):
		tmp = (x_m / (t - z)) * (y - z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if ((t_1 <= -4e+289) || !(t_1 <= 5e+303))
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m * (y - z)) / (t - z);
	tmp = 0.0;
	if ((t_1 <= -4e+289) || ~((t_1 <= 5e+303)))
		tmp = (x_m / (t - z)) * (y - z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[Or[LessEqual[t$95$1, -4e+289], N[Not[LessEqual[t$95$1, 5e+303]], $MachinePrecision]], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -4.0000000000000002e289 or 4.9999999999999997e303 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 49.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -4.0000000000000002e289 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 4.9999999999999997e303

   1. Initial program 98.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -4 \cdot 10^{+289} \lor \neg \left(\frac{x \cdot \left(y - z\right)}{t - z} \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-x\_m, \frac{y}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot y\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x\_m}{z}, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -5.2e-17)
    (fma (- x_m) (/ y z) x_m)
    (if (<= z 1.5e-81)
      (* (/ x_m (- t z)) y)
      (if (<= z 6.2e+54) (/ (* (- y z) x_m) t) (fma (- y) (/ x_m z) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -5.2e-17) {
		tmp = fma(-x_m, (y / z), x_m);
	} else if (z <= 1.5e-81) {
		tmp = (x_m / (t - z)) * y;
	} else if (z <= 6.2e+54) {
		tmp = ((y - z) * x_m) / t;
	} else {
		tmp = fma(-y, (x_m / z), x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -5.2e-17)
		tmp = fma(Float64(-x_m), Float64(y / z), x_m);
	elseif (z <= 1.5e-81)
		tmp = Float64(Float64(x_m / Float64(t - z)) * y);
	elseif (z <= 6.2e+54)
		tmp = Float64(Float64(Float64(y - z) * x_m) / t);
	else
		tmp = fma(Float64(-y), Float64(x_m / z), x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -5.2e-17], N[((-x$95$m) * N[(y / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 1.5e-81], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 6.2e+54], N[(N[(N[(y - z), $MachinePrecision] * x$95$m), $MachinePrecision] / t), $MachinePrecision], N[((-y) * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -5.20000000000000006e-17

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.2

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 80.4%

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

   if -5.20000000000000006e-17 < z < 1.4999999999999999e-81

   1. Initial program 93.8%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 85.1

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

   5. Applied rewrites 85.1%

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

   if 1.4999999999999999e-81 < z < 6.1999999999999999e54

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if 6.1999999999999999e54 < z

   1. Initial program 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 85.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -165000:\\ \;\;\;\;x\_m \cdot \left(1 - \frac{y - t}{z}\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+201}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t - z} \cdot \left(-x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -165000.0)
    (* x_m (- 1.0 (/ (- y t) z)))
    (if (<= z 6.6e+201)
      (* (/ x_m (- t z)) (- y z))
      (* (/ z (- t z)) (- x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -165000.0) {
		tmp = x_m * (1.0 - ((y - t) / z));
	} else if (z <= 6.6e+201) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = (z / (t - z)) * -x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-165000.0d0)) then
        tmp = x_m * (1.0d0 - ((y - t) / z))
    else if (z <= 6.6d+201) then
        tmp = (x_m / (t - z)) * (y - z)
    else
        tmp = (z / (t - z)) * -x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -165000.0) {
		tmp = x_m * (1.0 - ((y - t) / z));
	} else if (z <= 6.6e+201) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = (z / (t - z)) * -x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -165000.0:
		tmp = x_m * (1.0 - ((y - t) / z))
	elif z <= 6.6e+201:
		tmp = (x_m / (t - z)) * (y - z)
	else:
		tmp = (z / (t - z)) * -x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -165000.0)
		tmp = Float64(x_m * Float64(1.0 - Float64(Float64(y - t) / z)));
	elseif (z <= 6.6e+201)
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	else
		tmp = Float64(Float64(z / Float64(t - z)) * Float64(-x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -165000.0)
		tmp = x_m * (1.0 - ((y - t) / z));
	elseif (z <= 6.6e+201)
		tmp = (x_m / (t - z)) * (y - z);
	else
		tmp = (z / (t - z)) * -x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -165000.0], N[(x$95$m * N[(1.0 - N[(N[(y - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.6e+201], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] * (-x$95$m)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -165000

   1. Initial program 78.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 65.5

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

   4. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. difference-of-squares-rev N/A

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

      9. lift-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-/r/ N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. difference-of-squares-rev N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. *-commutative N/A

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

      20. lift--.f64 N/A

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

      21. associate-/l* N/A

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

      22. associate-*r* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 79.8

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

   9. Applied rewrites 79.8%

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

   if -165000 < z < 6.6e201

   1. Initial program 91.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.2

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

   4. Applied rewrites 91.2%

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

   if 6.6e201 < z

   1. Initial program 71.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 5: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-17} \lor \neg \left(z \leq 6.2 \cdot 10^{+54}\right):\\ \;\;\;\;\mathsf{fma}\left(-x\_m, \frac{y}{z}, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x\_m}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -5.2e-17) (not (<= z 6.2e+54)))
    (fma (- x_m) (/ y z) x_m)
    (/ (* (- y z) x_m) t))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -5.2e-17) || !(z <= 6.2e+54)) {
		tmp = fma(-x_m, (y / z), x_m);
	} else {
		tmp = ((y - z) * x_m) / t;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -5.2e-17) || !(z <= 6.2e+54))
		tmp = fma(Float64(-x_m), Float64(y / z), x_m);
	else
		tmp = Float64(Float64(Float64(y - z) * x_m) / t);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -5.2e-17], N[Not[LessEqual[z, 6.2e+54]], $MachinePrecision]], N[((-x$95$m) * N[(y / z), $MachinePrecision] + x$95$m), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] * x$95$m), $MachinePrecision] / t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000006e-17 or 6.1999999999999999e54 < z

   1. Initial program 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 68.1

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 77.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 82.6%

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

   if -5.20000000000000006e-17 < z < 6.1999999999999999e54

   1. Initial program 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 74.6

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

   7. Applied rewrites 74.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-17} \lor \neg \left(z \leq 6.2 \cdot 10^{+54}\right):\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-73} \lor \neg \left(z \leq 4.2 \cdot 10^{+54}\right):\\ \;\;\;\;\mathsf{fma}\left(-x\_m, \frac{y}{z}, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x\_m}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -3.9e-73) (not (<= z 4.2e+54)))
    (fma (- x_m) (/ y z) x_m)
    (/ (* y x_m) t))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -3.9e-73) || !(z <= 4.2e+54)) {
		tmp = fma(-x_m, (y / z), x_m);
	} else {
		tmp = (y * x_m) / t;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -3.9e-73) || !(z <= 4.2e+54))
		tmp = fma(Float64(-x_m), Float64(y / z), x_m);
	else
		tmp = Float64(Float64(y * x_m) / t);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -3.9e-73], N[Not[LessEqual[z, 4.2e+54]], $MachinePrecision]], N[((-x$95$m) * N[(y / z), $MachinePrecision] + x$95$m), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] / t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.89999999999999982e-73 or 4.19999999999999972e54 < z

   1. Initial program 78.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.9%

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

   if -3.89999999999999982e-73 < z < 4.19999999999999972e54

   1. Initial program 95.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-73} \lor \neg \left(z \leq 4.2 \cdot 10^{+54}\right):\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-17}:\\ \;\;\;\;x\_m \cdot \left(1 - \frac{y - t}{z}\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{y \cdot x\_m}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x\_m}{z}, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -5.8e-17)
    (* x_m (- 1.0 (/ (- y t) z)))
    (if (<= z 6.8e+54) (/ (* y x_m) (- t z)) (fma (- y) (/ x_m z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -5.8e-17) {
		tmp = x_m * (1.0 - ((y - t) / z));
	} else if (z <= 6.8e+54) {
		tmp = (y * x_m) / (t - z);
	} else {
		tmp = fma(-y, (x_m / z), x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -5.8e-17)
		tmp = Float64(x_m * Float64(1.0 - Float64(Float64(y - t) / z)));
	elseif (z <= 6.8e+54)
		tmp = Float64(Float64(y * x_m) / Float64(t - z));
	else
		tmp = fma(Float64(-y), Float64(x_m / z), x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -5.8e-17], N[(x$95$m * N[(1.0 - N[(N[(y - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.8e+54], N[(N[(y * x$95$m), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -5.8000000000000006e-17

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. difference-of-squares-rev N/A

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

      9. lift-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-/r/ N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. difference-of-squares-rev N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. *-commutative N/A

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

      20. lift--.f64 N/A

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

      21. associate-/l* N/A

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

      22. associate-*r* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in z around inf

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

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. associate-+l- N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 80.7

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

   9. Applied rewrites 80.7%

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

   if -5.8000000000000006e-17 < z < 6.8000000000000001e54

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if 6.8000000000000001e54 < z

   1. Initial program 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 85.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - \frac{y - t}{z}\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{y \cdot x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-x\_m, \frac{y}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{y \cdot x\_m}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x\_m}{z}, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -5.8e-17)
    (fma (- x_m) (/ y z) x_m)
    (if (<= z 6.8e+54) (/ (* y x_m) (- t z)) (fma (- y) (/ x_m z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -5.8e-17) {
		tmp = fma(-x_m, (y / z), x_m);
	} else if (z <= 6.8e+54) {
		tmp = (y * x_m) / (t - z);
	} else {
		tmp = fma(-y, (x_m / z), x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -5.8e-17)
		tmp = fma(Float64(-x_m), Float64(y / z), x_m);
	elseif (z <= 6.8e+54)
		tmp = Float64(Float64(y * x_m) / Float64(t - z));
	else
		tmp = fma(Float64(-y), Float64(x_m / z), x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -5.8e-17], N[((-x$95$m) * N[(y / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 6.8e+54], N[(N[(y * x$95$m), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -5.8000000000000006e-17

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.2

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 80.4%

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

   if -5.8000000000000006e-17 < z < 6.8000000000000001e54

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if 6.8000000000000001e54 < z

   1. Initial program 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 85.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{y \cdot x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-x\_m, \frac{y}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x\_m}{z}, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -5.2e-17)
    (fma (- x_m) (/ y z) x_m)
    (if (<= z 6.2e+54) (/ (* (- y z) x_m) t) (fma (- y) (/ x_m z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -5.2e-17) {
		tmp = fma(-x_m, (y / z), x_m);
	} else if (z <= 6.2e+54) {
		tmp = ((y - z) * x_m) / t;
	} else {
		tmp = fma(-y, (x_m / z), x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -5.2e-17)
		tmp = fma(Float64(-x_m), Float64(y / z), x_m);
	elseif (z <= 6.2e+54)
		tmp = Float64(Float64(Float64(y - z) * x_m) / t);
	else
		tmp = fma(Float64(-y), Float64(x_m / z), x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -5.2e-17], N[((-x$95$m) * N[(y / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 6.2e+54], N[(N[(N[(y - z), $MachinePrecision] * x$95$m), $MachinePrecision] / t), $MachinePrecision], N[((-y) * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -5.20000000000000006e-17

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.2

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 80.4%

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

   if -5.20000000000000006e-17 < z < 6.1999999999999999e54

   1. Initial program 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 74.6

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

   7. Applied rewrites 74.6%

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

   if 6.1999999999999999e54 < z

   1. Initial program 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 85.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -750 \lor \neg \left(z \leq 1.42 \cdot 10^{+54}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, t, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -750.0) (not (<= z 1.42e+54)))
    (fma (/ x_m z) t x_m)
    (* x_m (/ y t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -750.0) || !(z <= 1.42e+54)) {
		tmp = fma((x_m / z), t, x_m);
	} else {
		tmp = x_m * (y / t);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -750.0) || !(z <= 1.42e+54))
		tmp = fma(Float64(x_m / z), t, x_m);
	else
		tmp = Float64(x_m * Float64(y / t));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -750.0], N[Not[LessEqual[z, 1.42e+54]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * t + x$95$m), $MachinePrecision], N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -750 or 1.41999999999999995e54 < z

   1. Initial program 77.3%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.3%

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

   if -750 < z < 1.41999999999999995e54

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.9%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -750 \lor \neg \left(z \leq 1.42 \cdot 10^{+54}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \frac{y}{t}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* x_m (/ y t))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m * (y / t));
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m * (y / t))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m * (y / t));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * (x_m * (y / t))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m * Float64(y / t)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m * (y / t));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 41.4

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

5. Applied rewrites 41.4%

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

7. Applied rewrites 42.1%

   \[\leadsto x \cdot \color{blue}{\frac{y}{t}} \]
8. Add Preprocessing

Alternative 12: 6.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{x\_m}{z} \cdot t\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* (/ x_m z) t)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * ((x_m / z) * t);
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * ((x_m / z) * t)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * ((x_m / z) * t);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * ((x_m / z) * t)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(Float64(x_m / z) * t))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * ((x_m / z) * t);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(N[(x$95$m / z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 81.1

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

4. Applied rewrites 81.1%

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

5. Taylor expanded in z around inf

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

   1. associate--l+ N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. associate-*r/ N/A

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

   6. div-add-rev N/A

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

   7. mul-1-neg N/A

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

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

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. +-commutative N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

   16. lower-fma.f64 N/A

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

7. Applied rewrites 51.3%

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

8. Taylor expanded in t around inf

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

10. Applied rewrites 6.0%

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

11. Final simplification 6.0%

    \[\leadsto \frac{x}{z} \cdot t \]
12. Add Preprocessing

Developer Target 1: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{t - z}{y - z}} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024359 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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