Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A

Percentage Accurate: 98.4% → 98.4%

Time: 9.0s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

Alternative 2: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;x + y \cdot \frac{-t}{z - a}\\ \mathbf{elif}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(-t\right) \cdot y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -10.0)
     (+ x (* y (/ (- t) (- z a))))
     (if (<= t_1 1e-15)
       (fma (/ (- t z) a) y x)
       (if (<= t_1 2.0)
         (fma (/ (- z t) z) y x)
         (+ x (/ (* (- t) y) (- z a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -10.0) {
		tmp = x + (y * (-t / (z - a)));
	} else if (t_1 <= 1e-15) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = fma(((z - t) / z), y, x);
	} else {
		tmp = x + ((-t * y) / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = Float64(x + Float64(y * Float64(Float64(-t) / Float64(z - a))));
	elseif (t_1 <= 1e-15)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	else
		tmp = Float64(x + Float64(Float64(Float64(-t) * y) / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], N[(x + N[(y * N[((-t) / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-15], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(N[((-t) * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -10

   1. Initial program 96.3%

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

   2. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.5

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

   4. Applied rewrites 95.5%

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

   if -10 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000000000001e-15

   1. Initial program 99.3%

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

   2. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 91.1

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

   4. Applied rewrites 91.1%

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

   6. Applied rewrites 99.0%

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

   if 1.0000000000000001e-15 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 96.8

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

   4. Applied rewrites 96.8%

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

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 90.5

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.0

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

   6. Applied rewrites 90.0%

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

Alternative 3: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := x + y \cdot \frac{-t}{z - a}\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (+ x (* y (/ (- t) (- z a))))))
   (if (<= t_1 -10.0)
     t_2
     (if (<= t_1 1e-15)
       (fma (/ (- t z) a) y x)
       (if (<= t_1 2.0) (fma (/ (- z t) z) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = x + (y * (-t / (z - a)));
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_2;
	} else if (t_1 <= 1e-15) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = fma(((z - t) / z), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(x + Float64(y * Float64(Float64(-t) / Float64(z - a))))
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = t_2;
	elseif (t_1 <= 1e-15)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(y * N[((-t) / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], t$95$2, If[LessEqual[t$95$1, 1e-15], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -10 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.8%

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

   2. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if -10 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000000000001e-15

   1. Initial program 99.3%

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

   2. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 91.1

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

   4. Applied rewrites 91.1%

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

   6. Applied rewrites 99.0%

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

   if 1.0000000000000001e-15 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 96.8

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

   4. Applied rewrites 96.8%

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

Alternative 4: 87.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-15)
     (fma (/ (- t z) a) y x)
     (if (<= t_1 1e+63) (fma (/ (- z t) z) y x) (* (- t) (/ y (- z a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-15) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 1e+63) {
		tmp = fma(((z - t) / z), y, x);
	} else {
		tmp = -t * (y / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-15)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 1e+63)
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	else
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-15], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000000000001e-15

   1. Initial program 98.3%

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

   2. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 80.4

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

   4. Applied rewrites 80.4%

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

   6. Applied rewrites 86.0%

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

   if 1.0000000000000001e-15 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.5%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 73.7

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

   4. Applied rewrites 73.7%

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

Alternative 5: 87.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-15)
     (fma (/ (- t z) a) y x)
     (if (<= t_1 1e+63) (fma (/ z (- z a)) y x) (* (- t) (/ y (- z a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-15) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 1e+63) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = -t * (y / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-15)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 1e+63)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-15], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000000000001e-15

   1. Initial program 98.3%

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

   2. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 80.4

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

   4. Applied rewrites 80.4%

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

   6. Applied rewrites 86.0%

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

   if 1.0000000000000001e-15 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 92.6

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

   4. Applied rewrites 92.6%

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

   if 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.5%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 73.7

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

   4. Applied rewrites 73.7%

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

Alternative 6: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-219}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-219)
     (+ x (* y (/ t a)))
     (if (<= t_1 1e+63) (fma (/ z (- z a)) y x) (* (- t) (/ y (- z a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-219) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 1e+63) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = -t * (y / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-219)
		tmp = Float64(x + Float64(y * Float64(t / a)));
	elseif (t_1 <= 1e+63)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-219], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-219

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

   if 2.0000000000000001e-219 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 90.2

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

   4. Applied rewrites 90.2%

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

   if 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.5%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 73.7

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

   4. Applied rewrites 73.7%

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

Alternative 7: 82.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-219}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 0.9995:\\ \;\;\;\;\mathsf{fma}\left(z, t\_2, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ y (- z a))))
   (if (<= t_1 2e-219)
     (+ x (* y (/ t a)))
     (if (<= t_1 0.9995)
       (fma z t_2 x)
       (if (<= t_1 1e+63) (+ x y) (* (- t) t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y / (z - a);
	double tmp;
	if (t_1 <= 2e-219) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 0.9995) {
		tmp = fma(z, t_2, x);
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = -t * t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(y / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-219)
		tmp = Float64(x + Float64(y * Float64(t / a)));
	elseif (t_1 <= 0.9995)
		tmp = fma(z, t_2, x);
	elseif (t_1 <= 1e+63)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(-t) * t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-219], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9995], N[(z * t$95$2 + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(x + y), $MachinePrecision], N[((-t) * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-219

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

   if 2.0000000000000001e-219 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.99950000000000006

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 79.3

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

   6. Applied rewrites 79.3%

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

   if 0.99950000000000006 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 92.9%

      \[\leadsto x + \color{blue}{y} \]

   if 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.5%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 73.7

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

   4. Applied rewrites 73.7%

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

Alternative 8: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-219}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-28}:\\ \;\;\;\;x - \frac{y}{a} \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-219)
     (+ x (* y (/ t a)))
     (if (<= t_1 2e-28)
       (- x (* (/ y a) z))
       (if (<= t_1 1e+63) (+ x y) (* (- t) (/ y (- z a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-219) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 2e-28) {
		tmp = x - ((y / a) * z);
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = -t * (y / (z - a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    if (t_1 <= 2d-219) then
        tmp = x + (y * (t / a))
    else if (t_1 <= 2d-28) then
        tmp = x - ((y / a) * z)
    else if (t_1 <= 1d+63) then
        tmp = x + y
    else
        tmp = -t * (y / (z - a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-219) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 2e-28) {
		tmp = x - ((y / a) * z);
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = -t * (y / (z - a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if t_1 <= 2e-219:
		tmp = x + (y * (t / a))
	elif t_1 <= 2e-28:
		tmp = x - ((y / a) * z)
	elif t_1 <= 1e+63:
		tmp = x + y
	else:
		tmp = -t * (y / (z - a))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-219)
		tmp = Float64(x + Float64(y * Float64(t / a)));
	elseif (t_1 <= 2e-28)
		tmp = Float64(x - Float64(Float64(y / a) * z));
	elseif (t_1 <= 1e+63)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	tmp = 0.0;
	if (t_1 <= 2e-219)
		tmp = x + (y * (t / a));
	elseif (t_1 <= 2e-28)
		tmp = x - ((y / a) * z);
	elseif (t_1 <= 1e+63)
		tmp = x + y;
	else
		tmp = -t * (y / (z - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-219], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-28], N[(x - N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(x + y), $MachinePrecision], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-219

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

   if 2.0000000000000001e-219 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999994e-28

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.5

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

   7. Applied rewrites 80.5%

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

   if 1.99999999999999994e-28 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 90.5%

      \[\leadsto x + \color{blue}{y} \]

   if 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.5%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 73.7

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

   4. Applied rewrites 73.7%

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

Alternative 9: 81.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-219}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-28}:\\ \;\;\;\;x - \frac{y}{a} \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-219)
     (+ x (* y (/ t a)))
     (if (<= t_1 2e-28)
       (- x (* (/ y a) z))
       (if (<= t_1 1e+63) (+ x y) (fma (/ y a) t x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-219) {
		tmp = x + (y * (t / a));
	} else if (t_1 <= 2e-28) {
		tmp = x - ((y / a) * z);
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-219)
		tmp = Float64(x + Float64(y * Float64(t / a)));
	elseif (t_1 <= 2e-28)
		tmp = Float64(x - Float64(Float64(y / a) * z));
	elseif (t_1 <= 1e+63)
		tmp = Float64(x + y);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-219], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-28], N[(x - N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(x + y), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-219

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

   if 2.0000000000000001e-219 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999994e-28

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.5

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

   7. Applied rewrites 80.5%

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

   if 1.99999999999999994e-28 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 90.5%

      \[\leadsto x + \color{blue}{y} \]

   if 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 63.6

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

   4. Applied rewrites 63.6%

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

Alternative 10: 81.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-28}:\\ \;\;\;\;x - \frac{y}{a} \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-219)
     (fma (/ t a) y x)
     (if (<= t_1 2e-28)
       (- x (* (/ y a) z))
       (if (<= t_1 1e+63) (+ x y) (fma (/ y a) t x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-219) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2e-28) {
		tmp = x - ((y / a) * z);
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-219)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2e-28)
		tmp = Float64(x - Float64(Float64(y / a) * z));
	elseif (t_1 <= 1e+63)
		tmp = Float64(x + y);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-219], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-28], N[(x - N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(x + y), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-219

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 75.1

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

   6. Applied rewrites 75.1%

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

   if 2.0000000000000001e-219 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999994e-28

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.5

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

   7. Applied rewrites 80.5%

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

   if 1.99999999999999994e-28 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 90.5%

      \[\leadsto x + \color{blue}{y} \]

   if 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 63.6

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

   4. Applied rewrites 63.6%

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

Alternative 11: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-15)
     (fma (/ t a) y x)
     (if (<= t_1 1e+63) (+ x y) (fma (/ y a) t x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-15) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-15)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 1e+63)
		tmp = Float64(x + y);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-15], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(x + y), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000000000001e-15

   1. Initial program 98.3%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 76.8

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

   4. Applied rewrites 76.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.8

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

   6. Applied rewrites 76.8%

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

   if 1.0000000000000001e-15 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 91.3%

      \[\leadsto x + \color{blue}{y} \]

   if 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 63.6

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

   4. Applied rewrites 63.6%

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

Alternative 12: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{if}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ y a) t x)))
   (if (<= t_1 1e-15) t_2 (if (<= t_1 1e+63) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((y / a), t, x);
	double tmp;
	if (t_1 <= 1e-15) {
		tmp = t_2;
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(y / a), t, x)
	tmp = 0.0
	if (t_1 <= 1e-15)
		tmp = t_2;
	elseif (t_1 <= 1e+63)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-15], t$95$2, If[LessEqual[t$95$1, 1e+63], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000000000001e-15 or 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.4%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 74.4

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

   4. Applied rewrites 74.4%

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

   if 1.0000000000000001e-15 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 91.3%

      \[\leadsto x + \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 71.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := t \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+38}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* t (/ y a))))
   (if (<= t_1 -1e+38)
     t_2
     (if (<= t_1 5e-124) x (if (<= t_1 1e+63) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = t * (y / a);
	double tmp;
	if (t_1 <= -1e+38) {
		tmp = t_2;
	} else if (t_1 <= 5e-124) {
		tmp = x;
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = t * (y / a)
    if (t_1 <= (-1d+38)) then
        tmp = t_2
    else if (t_1 <= 5d-124) then
        tmp = x
    else if (t_1 <= 1d+63) then
        tmp = x + y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = t * (y / a);
	double tmp;
	if (t_1 <= -1e+38) {
		tmp = t_2;
	} else if (t_1 <= 5e-124) {
		tmp = x;
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = t * (y / a)
	tmp = 0
	if t_1 <= -1e+38:
		tmp = t_2
	elif t_1 <= 5e-124:
		tmp = x
	elif t_1 <= 1e+63:
		tmp = x + y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(t * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -1e+38)
		tmp = t_2;
	elseif (t_1 <= 5e-124)
		tmp = x;
	elseif (t_1 <= 1e+63)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = t * (y / a);
	tmp = 0.0;
	if (t_1 <= -1e+38)
		tmp = t_2;
	elseif (t_1 <= 5e-124)
		tmp = x;
	elseif (t_1 <= 1e+63)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+38], t$95$2, If[LessEqual[t$95$1, 5e-124], x, If[LessEqual[t$95$1, 1e+63], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -9.99999999999999977e37 or 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 93.6

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in t around inf

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

      1. *-commutative N/A

         \[\leadsto -1 \cdot \frac{t \cdot y}{z - a} \]

      2. associate-/l* N/A

         \[\leadsto -1 \cdot \frac{t \cdot y}{z - a} \]

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 69.8

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

   9. Applied rewrites 69.8%

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

   10. Taylor expanded in z around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 46.5

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

   12. Applied rewrites 46.5%

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

   if -9.99999999999999977e37 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.0000000000000003e-124

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.6%

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

   if 5.0000000000000003e-124 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 85.7%

      \[\leadsto x + \color{blue}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y \cdot t}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+38}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* y t) a)))
   (if (<= t_1 -1e+38)
     t_2
     (if (<= t_1 5e-124) x (if (<= t_1 1e+63) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y * t) / a;
	double tmp;
	if (t_1 <= -1e+38) {
		tmp = t_2;
	} else if (t_1 <= 5e-124) {
		tmp = x;
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = (y * t) / a
    if (t_1 <= (-1d+38)) then
        tmp = t_2
    else if (t_1 <= 5d-124) then
        tmp = x
    else if (t_1 <= 1d+63) then
        tmp = x + y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y * t) / a;
	double tmp;
	if (t_1 <= -1e+38) {
		tmp = t_2;
	} else if (t_1 <= 5e-124) {
		tmp = x;
	} else if (t_1 <= 1e+63) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = (y * t) / a
	tmp = 0
	if t_1 <= -1e+38:
		tmp = t_2
	elif t_1 <= 5e-124:
		tmp = x
	elif t_1 <= 1e+63:
		tmp = x + y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(y * t) / a)
	tmp = 0.0
	if (t_1 <= -1e+38)
		tmp = t_2;
	elseif (t_1 <= 5e-124)
		tmp = x;
	elseif (t_1 <= 1e+63)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = (y * t) / a;
	tmp = 0.0;
	if (t_1 <= -1e+38)
		tmp = t_2;
	elseif (t_1 <= 5e-124)
		tmp = x;
	elseif (t_1 <= 1e+63)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+38], t$95$2, If[LessEqual[t$95$1, 5e-124], x, If[LessEqual[t$95$1, 1e+63], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -9.99999999999999977e37 or 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.6%

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

   2. Taylor expanded in a around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 61.7

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

   4. Applied rewrites 61.7%

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

   5. Taylor expanded in t around inf

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 44.7

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

   7. Applied rewrites 44.7%

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

   if -9.99999999999999977e37 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.0000000000000003e-124

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.6%

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

   if 5.0000000000000003e-124 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 85.7%

      \[\leadsto x + \color{blue}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 1.15 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 1.15e-119) x (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 1.15e-119) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((z - t) / (z - a)) <= 1.15d-119) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 1.15e-119) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 1.15e-119:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 1.15e-119)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (z - a)) <= 1.15e-119)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 1.15e-119], x, N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.14999999999999997e-119

   1. Initial program 98.1%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 57.3%

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

   if 1.14999999999999997e-119 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 73.8%

      \[\leadsto x + \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-122}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.65e-87) x (if (<= x 7.5e-122) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.65e-87) {
		tmp = x;
	} else if (x <= 7.5e-122) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-1.65d-87)) then
        tmp = x
    else if (x <= 7.5d-122) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.65e-87) {
		tmp = x;
	} else if (x <= 7.5e-122) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.65e-87:
		tmp = x
	elif x <= 7.5e-122:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.65e-87)
		tmp = x;
	elseif (x <= 7.5e-122)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.65e-87)
		tmp = x;
	elseif (x <= 7.5e-122)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.65e-87], x, If[LessEqual[x, 7.5e-122], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e-87 or 7.4999999999999998e-122 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.6%

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

   if -1.65e-87 < x < 7.4999999999999998e-122

   1. Initial program 98.1%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 58.9

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 38.4

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

   7. Applied rewrites 38.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \]
   9. Step-by-step derivation

   10. Applied rewrites 29.4%

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

Alternative 17: 50.7% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y, z, t, a):
	return x

Julia

function code(x, y, z, t, a)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 50.7%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64
  (+ x (* y (/ (- z t) (- z a)))))