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

Percentage Accurate: 83.7% → 96.7%

Time: 7.3s

Alternatives: 8

Speedup: 0.8×

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: 83.7% 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: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

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 (<= x_m 5.2e-30)
    (/ (* x_m (- y z)) (- t z))
    (* (/ 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 tmp;
	if (x_m <= 5.2e-30) {
		tmp = (x_m * (y - z)) / (t - z);
	} 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) :: tmp
    if (x_m <= 5.2d-30) then
        tmp = (x_m * (y - z)) / (t - z)
    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 tmp;
	if (x_m <= 5.2e-30) {
		tmp = (x_m * (y - z)) / (t - z);
	} 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):
	tmp = 0
	if x_m <= 5.2e-30:
		tmp = (x_m * (y - z)) / (t - z)
	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)
	tmp = 0.0
	if (x_m <= 5.2e-30)
		tmp = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z));
	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)
	tmp = 0.0;
	if (x_m <= 5.2e-30)
		tmp = (x_m * (y - z)) / (t - z);
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 5.2e-30], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.19999999999999973e-30

   1. Initial program 87.1%

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

   if 5.19999999999999973e-30 < x

   1. Initial program 77.0%

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

   4. Applied rewrites 99.8%

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

Alternative 2: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+134} \lor \neg \left(z \leq 9.6 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{z}{t - z} \cdot \left(-x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

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 (or (<= z -7e+134) (not (<= z 9.6e+141)))
    (* (/ z (- t z)) (- x_m))
    (* (/ 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 tmp;
	if ((z <= -7e+134) || !(z <= 9.6e+141)) {
		tmp = (z / (t - z)) * -x_m;
	} 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) :: tmp
    if ((z <= (-7d+134)) .or. (.not. (z <= 9.6d+141))) then
        tmp = (z / (t - z)) * -x_m
    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 tmp;
	if ((z <= -7e+134) || !(z <= 9.6e+141)) {
		tmp = (z / (t - z)) * -x_m;
	} 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):
	tmp = 0
	if (z <= -7e+134) or not (z <= 9.6e+141):
		tmp = (z / (t - z)) * -x_m
	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)
	tmp = 0.0
	if ((z <= -7e+134) || !(z <= 9.6e+141))
		tmp = Float64(Float64(z / Float64(t - z)) * Float64(-x_m));
	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)
	tmp = 0.0;
	if ((z <= -7e+134) || ~((z <= 9.6e+141)))
		tmp = (z / (t - z)) * -x_m;
	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_] := N[(x$95$s * If[Or[LessEqual[z, -7e+134], N[Not[LessEqual[z, 9.6e+141]], $MachinePrecision]], N[(N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] * (-x$95$m)), $MachinePrecision], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.00000000000000006e134 or 9.59999999999999989e141 < z

   1. Initial program 64.3%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 91.5

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

   5. Applied rewrites 91.5%

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

   if -7.00000000000000006e134 < z < 9.59999999999999989e141

   1. Initial program 93.0%

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

   4. Applied rewrites 91.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+134} \lor \neg \left(z \leq 9.6 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{z}{t - z} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-46} \lor \neg \left(t \leq 8.8 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{y - z}{t} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m - \frac{x\_m}{z} \cdot y\\ \end{array} \end{array} \]

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 (<= t -2.2e-46) (not (<= t 8.8e-26)))
    (* (/ (- y z) t) x_m)
    (- x_m (* (/ x_m z) y)))))

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 ((t <= -2.2e-46) || !(t <= 8.8e-26)) {
		tmp = ((y - z) / t) * x_m;
	} else {
		tmp = x_m - ((x_m / z) * y);
	}
	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 ((t <= (-2.2d-46)) .or. (.not. (t <= 8.8d-26))) then
        tmp = ((y - z) / t) * x_m
    else
        tmp = x_m - ((x_m / z) * y)
    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 ((t <= -2.2e-46) || !(t <= 8.8e-26)) {
		tmp = ((y - z) / t) * x_m;
	} else {
		tmp = x_m - ((x_m / z) * y);
	}
	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 (t <= -2.2e-46) or not (t <= 8.8e-26):
		tmp = ((y - z) / t) * x_m
	else:
		tmp = x_m - ((x_m / z) * y)
	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 ((t <= -2.2e-46) || !(t <= 8.8e-26))
		tmp = Float64(Float64(Float64(y - z) / t) * x_m);
	else
		tmp = Float64(x_m - Float64(Float64(x_m / z) * y));
	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 ((t <= -2.2e-46) || ~((t <= 8.8e-26)))
		tmp = ((y - z) / t) * x_m;
	else
		tmp = x_m - ((x_m / z) * y);
	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[Or[LessEqual[t, -2.2e-46], N[Not[LessEqual[t, 8.8e-26]], $MachinePrecision]], N[(N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] * x$95$m), $MachinePrecision], N[(x$95$m - N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -2.2000000000000001e-46 or 8.8000000000000003e-26 < t

   1. Initial program 83.6%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.4

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

   5. Applied rewrites 77.4%

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

   if -2.2000000000000001e-46 < t < 8.8000000000000003e-26

   1. Initial program 85.5%

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 88.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-46} \lor \neg \left(t \leq 8.8 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{y - z}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-33} \lor \neg \left(z \leq 38\right):\\ \;\;\;\;x\_m - \frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot y\\ \end{array} \end{array} \]

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 -3.1e-33) (not (<= z 38.0)))
    (- x_m (* (/ x_m z) y))
    (* (/ x_m (- t z)) y))))

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 <= -3.1e-33) || !(z <= 38.0)) {
		tmp = x_m - ((x_m / z) * y);
	} else {
		tmp = (x_m / (t - z)) * y;
	}
	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 <= (-3.1d-33)) .or. (.not. (z <= 38.0d0))) then
        tmp = x_m - ((x_m / z) * y)
    else
        tmp = (x_m / (t - z)) * y
    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 <= -3.1e-33) || !(z <= 38.0)) {
		tmp = x_m - ((x_m / z) * y);
	} else {
		tmp = (x_m / (t - z)) * y;
	}
	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 <= -3.1e-33) or not (z <= 38.0):
		tmp = x_m - ((x_m / z) * y)
	else:
		tmp = (x_m / (t - z)) * y
	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 <= -3.1e-33) || !(z <= 38.0))
		tmp = Float64(x_m - Float64(Float64(x_m / z) * y));
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) * y);
	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 <= -3.1e-33) || ~((z <= 38.0)))
		tmp = x_m - ((x_m / z) * y);
	else
		tmp = (x_m / (t - z)) * y;
	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[Or[LessEqual[z, -3.1e-33], N[Not[LessEqual[z, 38.0]], $MachinePrecision]], N[(x$95$m - N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.09999999999999997e-33 or 38 < z

   1. Initial program 75.9%

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 58.1

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

   7. Applied rewrites 58.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 70.2%

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

   if -3.09999999999999997e-33 < z < 38

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{t - z}} \]
   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 76.8

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

   5. Applied rewrites 76.8%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-33} \lor \neg \left(z \leq 38\right):\\ \;\;\;\;x - \frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-36} \lor \neg \left(z \leq 5.5\right):\\ \;\;\;\;x\_m - \frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\_m\\ \end{array} \end{array} \]

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 -8.8e-36) (not (<= z 5.5)))
    (- x_m (* (/ x_m z) y))
    (* (/ y t) 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 <= -8.8e-36) || !(z <= 5.5)) {
		tmp = x_m - ((x_m / z) * y);
	} else {
		tmp = (y / t) * 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 <= (-8.8d-36)) .or. (.not. (z <= 5.5d0))) then
        tmp = x_m - ((x_m / z) * y)
    else
        tmp = (y / t) * 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 <= -8.8e-36) || !(z <= 5.5)) {
		tmp = x_m - ((x_m / z) * y);
	} else {
		tmp = (y / t) * 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 <= -8.8e-36) or not (z <= 5.5):
		tmp = x_m - ((x_m / z) * y)
	else:
		tmp = (y / t) * 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 <= -8.8e-36) || !(z <= 5.5))
		tmp = Float64(x_m - Float64(Float64(x_m / z) * y));
	else
		tmp = Float64(Float64(y / t) * 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 <= -8.8e-36) || ~((z <= 5.5)))
		tmp = x_m - ((x_m / z) * y);
	else
		tmp = (y / t) * 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[Or[LessEqual[z, -8.8e-36], N[Not[LessEqual[z, 5.5]], $MachinePrecision]], N[(x$95$m - N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -8.7999999999999997e-36 or 5.5 < z

   1. Initial program 75.9%

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 58.1

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

   7. Applied rewrites 58.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 70.2%

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

   if -8.7999999999999997e-36 < z < 5.5

   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 68.3

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 69.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-36} \lor \neg \left(z \leq 5.5\right):\\ \;\;\;\;x - \frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+49} \lor \neg \left(z \leq 800\right):\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\_m\\ \end{array} \end{array} \]

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 -4.5e+49) (not (<= z 800.0))) (* 1.0 x_m) (* (/ y t) 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 <= -4.5e+49) || !(z <= 800.0)) {
		tmp = 1.0 * x_m;
	} else {
		tmp = (y / t) * 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 <= (-4.5d+49)) .or. (.not. (z <= 800.0d0))) then
        tmp = 1.0d0 * x_m
    else
        tmp = (y / t) * 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 <= -4.5e+49) || !(z <= 800.0)) {
		tmp = 1.0 * x_m;
	} else {
		tmp = (y / t) * 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 <= -4.5e+49) or not (z <= 800.0):
		tmp = 1.0 * x_m
	else:
		tmp = (y / t) * 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 <= -4.5e+49) || !(z <= 800.0))
		tmp = Float64(1.0 * x_m);
	else
		tmp = Float64(Float64(y / t) * 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 <= -4.5e+49) || ~((z <= 800.0)))
		tmp = 1.0 * x_m;
	else
		tmp = (y / t) * 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[Or[LessEqual[z, -4.5e+49], N[Not[LessEqual[z, 800.0]], $MachinePrecision]], N[(1.0 * x$95$m), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.49999999999999982e49 or 800 < z

   1. Initial program 72.3%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.3

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

   7. Applied rewrites 81.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.5%

       \[\leadsto \color{blue}{1} \cdot x \]

   if -4.49999999999999982e49 < z < 800

   1. Initial program 95.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 63.9

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

   5. Applied rewrites 63.9%

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

   7. Applied rewrites 64.6%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+49} \lor \neg \left(z \leq 800\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+49} \lor \neg \left(z \leq 800\right):\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t} \cdot y\\ \end{array} \end{array} \]

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 -4.5e+49) (not (<= z 800.0))) (* 1.0 x_m) (* (/ x_m t) y))))

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 <= -4.5e+49) || !(z <= 800.0)) {
		tmp = 1.0 * x_m;
	} else {
		tmp = (x_m / t) * y;
	}
	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 <= (-4.5d+49)) .or. (.not. (z <= 800.0d0))) then
        tmp = 1.0d0 * x_m
    else
        tmp = (x_m / t) * y
    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 <= -4.5e+49) || !(z <= 800.0)) {
		tmp = 1.0 * x_m;
	} else {
		tmp = (x_m / t) * y;
	}
	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 <= -4.5e+49) or not (z <= 800.0):
		tmp = 1.0 * x_m
	else:
		tmp = (x_m / t) * y
	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 <= -4.5e+49) || !(z <= 800.0))
		tmp = Float64(1.0 * x_m);
	else
		tmp = Float64(Float64(x_m / t) * y);
	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 <= -4.5e+49) || ~((z <= 800.0)))
		tmp = 1.0 * x_m;
	else
		tmp = (x_m / t) * y;
	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[Or[LessEqual[z, -4.5e+49], N[Not[LessEqual[z, 800.0]], $MachinePrecision]], N[(1.0 * x$95$m), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.49999999999999982e49 or 800 < z

   1. Initial program 72.3%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.3

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

   7. Applied rewrites 81.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.5%

       \[\leadsto \color{blue}{1} \cdot x \]

   if -4.49999999999999982e49 < z < 800

   1. Initial program 95.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 63.9

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

   5. Applied rewrites 63.9%

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

   7. Applied rewrites 58.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+49} \lor \neg \left(z \leq 800\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.2% accurate, 3.8× speedup

Math

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

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 (* 1.0 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) {
	return x_s * (1.0 * x_m);
}

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 * (1.0d0 * x_m)
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 * (1.0 * x_m);
}

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 * (1.0 * x_m)

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(1.0 * x_m))
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 * (1.0 * x_m);
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[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.4%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 43.7

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

5. Applied rewrites 43.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 54.4

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

7. Applied rewrites 54.4%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 34.4%

    \[\leadsto \color{blue}{1} \cdot x \]
11. Add Preprocessing

Developer Target 1: 97.0% 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 2025008 
(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)))