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

Percentage Accurate: 98.1% → 98.1%

Time: 3.6s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

Alternative 2: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{-t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z (- a t)) y x)))
   (if (<= t_1 -50000000.0)
     t_2
     (if (<= t_1 5e-24)
       (+ x (* y (/ (- z t) a)))
       (if (<= t_1 2.0) (fma (/ (- z t) (- t)) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / (a - t)), y, x);
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-24) {
		tmp = x + (y * ((z - t) / a));
	} else if (t_1 <= 2.0) {
		tmp = fma(((z - t) / -t), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / Float64(a - t)), y, x)
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-24)
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / a)));
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(Float64(z - t) / Float64(-t)), 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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, 5e-24], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(N[(z - t), $MachinePrecision] / (-t)), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5e7 or 2 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 94.5%

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

   if -5e7 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 98.1%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 96.5

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

   7. Applied rewrites 96.5%

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

Alternative 3: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z (- a t)) y x)))
   (if (<= t_1 -50000000.0)
     t_2
     (if (<= t_1 5e-24)
       (+ x (* y (/ (- z t) a)))
       (if (<= t_1 2.0) (fma (/ (- t) (- a t)) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / (a - t)), y, x);
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-24) {
		tmp = x + (y * ((z - t) / a));
	} else if (t_1 <= 2.0) {
		tmp = fma((-t / (a - t)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / Float64(a - t)), y, x)
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-24)
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / a)));
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(Float64(-t) / Float64(a - t)), 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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, 5e-24], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[((-t) / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5e7 or 2 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 94.5%

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

   if -5e7 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 98.1%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.3

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

   7. Applied rewrites 98.3%

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

Alternative 4: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z (- a t)) y x)))
   (if (<= t_1 -50000000.0)
     t_2
     (if (<= t_1 5e-24)
       (+ x (* y (/ (- z t) a)))
       (if (<= t_1 2.0) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / (a - t)), y, x);
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-24) {
		tmp = x + (y * ((z - t) / a));
	} else if (t_1 <= 2.0) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / Float64(a - t)), y, x)
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-24)
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / a)));
	elseif (t_1 <= 2.0)
		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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, 5e-24], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5e7 or 2 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 94.5%

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

   if -5e7 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 98.1%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.5%

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

Alternative 5: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z (- a t)) y x)))
   (if (<= t_1 -50000000.0)
     t_2
     (if (<= t_1 5e-24)
       (fma y (/ (- z t) a) x)
       (if (<= t_1 2.0) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / (a - t)), y, x);
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-24) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 2.0) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / Float64(a - t)), y, x)
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-24)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 2.0)
		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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, 5e-24], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5e7 or 2 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 94.5%

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

   if -5e7 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24

   1. Initial program 99.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.5%

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

Alternative 6: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{-t}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 100:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -1e+125)
     (fma (/ z (- t)) y x)
     (if (<= t_1 5e-24)
       (fma y (/ (- z t) a) x)
       (if (<= t_1 100.0) (+ x y) (* y (/ z (- a t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1e+125) {
		tmp = fma((z / -t), y, x);
	} else if (t_1 <= 5e-24) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 100.0) {
		tmp = x + y;
	} else {
		tmp = y * (z / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1e+125)
		tmp = fma(Float64(z / Float64(-t)), y, x);
	elseif (t_1 <= 5e-24)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 100.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(z / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+125], N[(N[(z / (-t)), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e-24], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 100.0], N[(x + y), $MachinePrecision], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -9.9999999999999992e124

   1. Initial program 90.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 56.1

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

   10. Applied rewrites 56.1%

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

   if -9.9999999999999992e124 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24

   1. Initial program 99.3%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 92.0

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

   5. Applied rewrites 92.0%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 100

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.3%

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

   if 100 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 7: 81.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{-t}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 100:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -1e+125)
     (fma (/ z (- t)) y x)
     (if (<= t_1 5e-24)
       (fma y (/ z a) x)
       (if (<= t_1 100.0) (+ x y) (* y (/ z (- a t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1e+125) {
		tmp = fma((z / -t), y, x);
	} else if (t_1 <= 5e-24) {
		tmp = fma(y, (z / a), x);
	} else if (t_1 <= 100.0) {
		tmp = x + y;
	} else {
		tmp = y * (z / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1e+125)
		tmp = fma(Float64(z / Float64(-t)), y, x);
	elseif (t_1 <= 5e-24)
		tmp = fma(y, Float64(z / a), x);
	elseif (t_1 <= 100.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(z / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+125], N[(N[(z / (-t)), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e-24], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 100.0], N[(x + y), $MachinePrecision], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -9.9999999999999992e124

   1. Initial program 90.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 56.1

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

   10. Applied rewrites 56.1%

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

   if -9.9999999999999992e124 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 100

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.3%

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

   if 100 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 8: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := y \cdot \frac{z}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+83}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 100:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (* y (/ z (- a t)))))
   (if (<= t_1 -5e+83)
     t_2
     (if (<= t_1 5e-24) (fma y (/ z a) x) (if (<= t_1 100.0) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = y * (z / (a - t));
	double tmp;
	if (t_1 <= -5e+83) {
		tmp = t_2;
	} else if (t_1 <= 5e-24) {
		tmp = fma(y, (z / a), x);
	} else if (t_1 <= 100.0) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(y * Float64(z / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -5e+83)
		tmp = t_2;
	elseif (t_1 <= 5e-24)
		tmp = fma(y, Float64(z / a), x);
	elseif (t_1 <= 100.0)
		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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+83], t$95$2, If[LessEqual[t$95$1, 5e-24], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 100.0], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5.00000000000000029e83 or 100 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if -5.00000000000000029e83 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 100

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.3%

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

Alternative 9: 71.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+228}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 10^{+69}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -2e+228)
     (/ (* y z) a)
     (if (<= t_1 5e-30) x (if (<= t_1 1e+69) (+ x y) (* y (/ z a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -2e+228) {
		tmp = (y * z) / a;
	} else if (t_1 <= 5e-30) {
		tmp = x;
	} else if (t_1 <= 1e+69) {
		tmp = x + y;
	} else {
		tmp = 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) / (a - t)
    if (t_1 <= (-2d+228)) then
        tmp = (y * z) / a
    else if (t_1 <= 5d-30) then
        tmp = x
    else if (t_1 <= 1d+69) then
        tmp = x + y
    else
        tmp = 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) / (a - t);
	double tmp;
	if (t_1 <= -2e+228) {
		tmp = (y * z) / a;
	} else if (t_1 <= 5e-30) {
		tmp = x;
	} else if (t_1 <= 1e+69) {
		tmp = x + y;
	} else {
		tmp = y * (z / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	tmp = 0
	if t_1 <= -2e+228:
		tmp = (y * z) / a
	elif t_1 <= 5e-30:
		tmp = x
	elif t_1 <= 1e+69:
		tmp = x + y
	else:
		tmp = y * (z / a)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -2e+228)
		tmp = Float64(Float64(y * z) / a);
	elseif (t_1 <= 5e-30)
		tmp = x;
	elseif (t_1 <= 1e+69)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(z / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	tmp = 0.0;
	if (t_1 <= -2e+228)
		tmp = (y * z) / a;
	elseif (t_1 <= 5e-30)
		tmp = x;
	elseif (t_1 <= 1e+69)
		tmp = x + y;
	else
		tmp = 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[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+228], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 5e-30], x, If[LessEqual[t$95$1, 1e+69], N[(x + y), $MachinePrecision], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.9999999999999998e228

   1. Initial program 80.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in t around 0

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 54.4

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

   8. Applied rewrites 54.4%

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

   if -1.9999999999999998e228 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.99999999999999972e-30

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 61.6%

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

   if 4.99999999999999972e-30 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.0000000000000001e69

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 90.8%

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

   if 1.0000000000000001e69 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 93.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 42.4%

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

Alternative 10: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \frac{y \cdot z}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+228}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 10^{+131}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (/ (* y z) a)))
   (if (<= t_1 -2e+228)
     t_2
     (if (<= t_1 5e-30) x (if (<= t_1 1e+131) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (y * z) / a;
	double tmp;
	if (t_1 <= -2e+228) {
		tmp = t_2;
	} else if (t_1 <= 5e-30) {
		tmp = x;
	} else if (t_1 <= 1e+131) {
		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) / (a - t)
    t_2 = (y * z) / a
    if (t_1 <= (-2d+228)) then
        tmp = t_2
    else if (t_1 <= 5d-30) then
        tmp = x
    else if (t_1 <= 1d+131) 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) / (a - t);
	double t_2 = (y * z) / a;
	double tmp;
	if (t_1 <= -2e+228) {
		tmp = t_2;
	} else if (t_1 <= 5e-30) {
		tmp = x;
	} else if (t_1 <= 1e+131) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	t_2 = (y * z) / a
	tmp = 0
	if t_1 <= -2e+228:
		tmp = t_2
	elif t_1 <= 5e-30:
		tmp = x
	elif t_1 <= 1e+131:
		tmp = x + y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(Float64(y * z) / a)
	tmp = 0.0
	if (t_1 <= -2e+228)
		tmp = t_2;
	elseif (t_1 <= 5e-30)
		tmp = x;
	elseif (t_1 <= 1e+131)
		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) / (a - t);
	t_2 = (y * z) / a;
	tmp = 0.0;
	if (t_1 <= -2e+228)
		tmp = t_2;
	elseif (t_1 <= 5e-30)
		tmp = x;
	elseif (t_1 <= 1e+131)
		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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+228], t$95$2, If[LessEqual[t$95$1, 5e-30], x, If[LessEqual[t$95$1, 1e+131], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.9999999999999998e228 or 9.9999999999999991e130 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 87.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in t around 0

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 49.3

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

   8. Applied rewrites 49.3%

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

   if -1.9999999999999998e228 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.99999999999999972e-30

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 61.6%

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

   if 4.99999999999999972e-30 < (/.f64 (-.f64 z t) (-.f64 a t)) < 9.9999999999999991e130

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 86.5%

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

Alternative 11: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma y (/ z a) x)))
   (if (<= t_1 5e-24) t_2 (if (<= t_1 2.0) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma(y, (z / a), x);
	double tmp;
	if (t_1 <= 5e-24) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(y, Float64(z / a), x)
	tmp = 0.0
	if (t_1 <= 5e-24)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-24], t$95$2, If[LessEqual[t$95$1, 2.0], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e-24 or 2 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 4.9999999999999998e-24 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 96.5%

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

Alternative 12: 56.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+118}:\\ \;\;\;\;y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+181}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= t_1 -2e+118) y (if (<= t_1 5e+181) x y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t_1 <= -2e+118) {
		tmp = y;
	} else if (t_1 <= 5e+181) {
		tmp = x;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_1 = y * ((z - t) / (a - t))
    if (t_1 <= (-2d+118)) then
        tmp = y
    else if (t_1 <= 5d+181) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t_1 <= -2e+118) {
		tmp = y;
	} else if (t_1 <= 5e+181) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (a - t))
	tmp = 0
	if t_1 <= -2e+118:
		tmp = y
	elif t_1 <= 5e+181:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -2e+118)
		tmp = y;
	elseif (t_1 <= 5e+181)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (a - t));
	tmp = 0.0;
	if (t_1 <= -2e+118)
		tmp = y;
	elseif (t_1 <= 5e+181)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+118], y, If[LessEqual[t$95$1, 5e+181], x, y]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < -1.99999999999999993e118 or 5.0000000000000003e181 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in t around inf

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

      1. associate-*l/ 27.9

         \[\leadsto y \]

      2. *-commutative 27.9

         \[\leadsto y \]

   8. Applied rewrites 27.9%

      \[\leadsto y \]

   if -1.99999999999999993e118 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < 5.0000000000000003e181

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 67.1%

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

Alternative 13: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 1.15 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- a t)) 1.15e-27) x (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= 1.15e-27) {
		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) / (a - t)) <= 1.15d-27) 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) / (a - t)) <= 1.15e-27) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (a - t)) <= 1.15e-27:
		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(a - t)) <= 1.15e-27)
		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) / (a - t)) <= 1.15e-27)
		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[(a - t), $MachinePrecision]), $MachinePrecision], 1.15e-27], x, N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 1.15e-27

   1. Initial program 97.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 57.2%

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

   if 1.15e-27 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 98.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 77.7%

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

Alternative 14: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift--.f64 98.1

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

4. Applied rewrites 98.1%

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

Alternative 15: 51.6% accurate, 26.0× 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 98.1%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 51.6%

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

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (< y -8.508084860551241e-17)
     t_1
     (if (< y 2.894426862792089e-49)
       (+ x (* (* y (- z t)) (/ 1.0 (- a t))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * ((z - t) / (a - t)))
    if (y < (-8.508084860551241d-17)) then
        tmp = t_1
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) * (1.0d0 / (a - t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	tmp = 0
	if y < -8.508084860551241e-17:
		tmp = t_1
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) * Float64(1.0 / Float64(a - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -8.508084860551241e-17], t$95$1, If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2025089 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8508084860551241/100000000000000000000000000000000) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t)))))))

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