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

Percentage Accurate: 98.2% → 98.2%

Time: 9.6s

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: 98.2% 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: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

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

   3. associate-*r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. lower-fma.f64 N/A

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

   10. difference-of-squares N/A

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

   11. lift--.f64 N/A

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

   12. times-frac N/A

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

   13. lift-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 88.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

6. Applied rewrites 97.3%

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

Alternative 2: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \left(-t\right) \cdot \frac{y}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+155}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* (- t) (/ y (- z a)))))
   (if (<= t_1 -1e+155)
     t_2
     (if (<= t_1 1e-161)
       (fma (/ y a) t x)
       (if (<= t_1 1e+124) (fma (/ z (- z a)) 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 = -t * (y / (z - a));
	double tmp;
	if (t_1 <= -1e+155) {
		tmp = t_2;
	} else if (t_1 <= 1e-161) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 1e+124) {
		tmp = fma((z / (z - a)), 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(Float64(-t) * Float64(y / Float64(z - a)))
	tmp = 0.0
	if (t_1 <= -1e+155)
		tmp = t_2;
	elseif (t_1 <= 1e-161)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 1e+124)
		tmp = fma(Float64(z / Float64(z - a)), 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[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+155], t$95$2, If[LessEqual[t$95$1, 1e-161], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+124], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.00000000000000001e155 or 9.99999999999999948e123 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 90.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 18.3

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

   5. Applied rewrites 18.3%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 81.9

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

   8. Applied rewrites 81.9%

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

   if -1.00000000000000001e155 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000003e-161

   1. Initial program 97.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if 1.00000000000000003e-161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.99999999999999948e123

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

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

Alternative 3: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+114} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+153}\right):\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (or (<= t_1 -2e+114) (not (<= t_1 5e+153)))
     (* (- z t) (/ y (- z a)))
     (fma (/ z (- z a)) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if ((t_1 <= -2e+114) || !(t_1 <= 5e+153)) {
		tmp = (z - t) * (y / (z - a));
	} else {
		tmp = fma((z / (z - a)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if ((t_1 <= -2e+114) || !(t_1 <= 5e+153))
		tmp = Float64(Float64(z - t) * Float64(y / Float64(z - a)));
	else
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+114], N[Not[LessEqual[t$95$1, 5e+153]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -2e114 or 5.00000000000000018e153 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 94.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if -2e114 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 5.00000000000000018e153

   1. Initial program 98.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{z - a} \leq -2 \cdot 10^{+114} \lor \neg \left(y \cdot \frac{z - t}{z - a} \leq 5 \cdot 10^{+153}\right):\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+246} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (or (<= t_1 -4e+246) (not (<= t_1 2e+304))) (* (/ t a) y) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if ((t_1 <= -4e+246) || !(t_1 <= 2e+304)) {
		tmp = (t / a) * y;
	} else {
		tmp = y + 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) :: t_1
    real(8) :: tmp
    t_1 = y * ((z - t) / (z - a))
    if ((t_1 <= (-4d+246)) .or. (.not. (t_1 <= 2d+304))) then
        tmp = (t / a) * y
    else
        tmp = y + x
    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) / (z - a));
	double tmp;
	if ((t_1 <= -4e+246) || !(t_1 <= 2e+304)) {
		tmp = (t / a) * y;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (z - a))
	tmp = 0
	if (t_1 <= -4e+246) or not (t_1 <= 2e+304):
		tmp = (t / a) * y
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if ((t_1 <= -4e+246) || !(t_1 <= 2e+304))
		tmp = Float64(Float64(t / a) * y);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (z - a));
	tmp = 0.0;
	if ((t_1 <= -4e+246) || ~((t_1 <= 2e+304)))
		tmp = (t / a) * y;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+246], N[Not[LessEqual[t$95$1, 2e+304]], $MachinePrecision]], N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -4.00000000000000027e246 or 1.9999999999999999e304 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 87.1%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 45.1%

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

   12. Applied rewrites 55.3%

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

   if -4.00000000000000027e246 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 1.9999999999999999e304

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.4

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

   5. Applied rewrites 70.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{z - a} \leq -4 \cdot 10^{+246} \lor \neg \left(y \cdot \frac{z - t}{z - a} \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+246}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (<= t_1 -4e+246)
     (* (/ t a) y)
     (if (<= t_1 2e+304) (+ y x) (/ (* y t) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -4e+246) {
		tmp = (t / a) * y;
	} else if (t_1 <= 2e+304) {
		tmp = y + x;
	} else {
		tmp = (y * t) / 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 = y * ((z - t) / (z - a))
    if (t_1 <= (-4d+246)) then
        tmp = (t / a) * y
    else if (t_1 <= 2d+304) then
        tmp = y + x
    else
        tmp = (y * t) / 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 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -4e+246) {
		tmp = (t / a) * y;
	} else if (t_1 <= 2e+304) {
		tmp = y + x;
	} else {
		tmp = (y * t) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (z - a))
	tmp = 0
	if t_1 <= -4e+246:
		tmp = (t / a) * y
	elif t_1 <= 2e+304:
		tmp = y + x
	else:
		tmp = (y * t) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if (t_1 <= -4e+246)
		tmp = Float64(Float64(t / a) * y);
	elseif (t_1 <= 2e+304)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y * t) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (z - a));
	tmp = 0.0;
	if (t_1 <= -4e+246)
		tmp = (t / a) * y;
	elseif (t_1 <= 2e+304)
		tmp = y + x;
	else
		tmp = (y * t) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+246], N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+304], N[(y + x), $MachinePrecision], N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -4.00000000000000027e246

   1. Initial program 90.7%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 32.0%

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

   12. Applied rewrites 53.8%

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

   if -4.00000000000000027e246 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 1.9999999999999999e304

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if 1.9999999999999999e304 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 82.4%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 62.2

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

   7. Applied rewrites 62.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{z - a} \leq -4 \cdot 10^{+246}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{elif}\;y \cdot \frac{z - t}{z - a} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+246}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (<= t_1 -4e+246)
     (* (/ t a) y)
     (if (<= t_1 2e+304) (+ y x) (* (/ y a) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -4e+246) {
		tmp = (t / a) * y;
	} else if (t_1 <= 2e+304) {
		tmp = y + x;
	} else {
		tmp = (y / a) * t;
	}
	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) / (z - a))
    if (t_1 <= (-4d+246)) then
        tmp = (t / a) * y
    else if (t_1 <= 2d+304) then
        tmp = y + x
    else
        tmp = (y / a) * t
    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) / (z - a));
	double tmp;
	if (t_1 <= -4e+246) {
		tmp = (t / a) * y;
	} else if (t_1 <= 2e+304) {
		tmp = y + x;
	} else {
		tmp = (y / a) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (z - a))
	tmp = 0
	if t_1 <= -4e+246:
		tmp = (t / a) * y
	elif t_1 <= 2e+304:
		tmp = y + x
	else:
		tmp = (y / a) * t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if (t_1 <= -4e+246)
		tmp = Float64(Float64(t / a) * y);
	elseif (t_1 <= 2e+304)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / a) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (z - a));
	tmp = 0.0;
	if (t_1 <= -4e+246)
		tmp = (t / a) * y;
	elseif (t_1 <= 2e+304)
		tmp = y + x;
	else
		tmp = (y / a) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+246], N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+304], N[(y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -4.00000000000000027e246

   1. Initial program 90.7%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 32.0%

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

   12. Applied rewrites 53.8%

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

   if -4.00000000000000027e246 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 1.9999999999999999e304

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if 1.9999999999999999e304 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 82.4%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 62.2

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

   7. Applied rewrites 62.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.2%

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

   12. Applied rewrites 62.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{z - a} \leq -4 \cdot 10^{+246}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{elif}\;y \cdot \frac{z - t}{z - a} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;t\_1 \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t - a}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+155)
     (* (- t) (/ y (- z a)))
     (if (<= t_1 0.0004)
       (fma (- y) (/ (- z t) a) x)
       (+ x (* y (- 1.0 (/ (- t a) z))))))))

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+155) {
		tmp = -t * (y / (z - a));
	} else if (t_1 <= 0.0004) {
		tmp = fma(-y, ((z - t) / a), x);
	} else {
		tmp = x + (y * (1.0 - ((t - a) / z)));
	}
	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+155)
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	elseif (t_1 <= 0.0004)
		tmp = fma(Float64(-y), Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - Float64(Float64(t - a) / z))));
	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+155], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0004], N[((-y) * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[(N[(t - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.00000000000000001e155

   1. Initial program 89.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 18.4

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 79.4

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

   8. Applied rewrites 79.4%

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

   if -1.00000000000000001e155 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000019e-4

   1. Initial program 97.9%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 91.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 89.9

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

   9. Applied rewrites 89.9%

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

   if 4.00000000000000019e-4 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 87.9

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

   5. Applied rewrites 87.9%

      \[\leadsto x + y \cdot \color{blue}{\left(1 - \frac{t - a}{z}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t - a}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;t\_1 \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+155)
     (* (- t) (/ y (- z a)))
     (if (<= t_1 0.0004)
       (fma (- y) (/ (- z t) a) x)
       (+ x (* y (- 1.0 (/ t z))))))))

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+155) {
		tmp = -t * (y / (z - a));
	} else if (t_1 <= 0.0004) {
		tmp = fma(-y, ((z - t) / a), x);
	} else {
		tmp = x + (y * (1.0 - (t / z)));
	}
	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+155)
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	elseif (t_1 <= 0.0004)
		tmp = fma(Float64(-y), Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - Float64(t / z))));
	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+155], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0004], N[((-y) * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.00000000000000001e155

   1. Initial program 89.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 18.4

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 79.4

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

   8. Applied rewrites 79.4%

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

   if -1.00000000000000001e155 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000019e-4

   1. Initial program 97.9%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 91.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 89.9

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

   9. Applied rewrites 89.9%

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

   if 4.00000000000000019e-4 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 87.8%

      \[\leadsto x + y \cdot \left(1 - \color{blue}{\frac{t}{z}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;t\_1 \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+155)
     (* (- t) (/ y (- z a)))
     (if (<= t_1 0.0004)
       (fma (- y) (/ (- z t) a) x)
       (fma (/ (- z t) z) y 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+155) {
		tmp = -t * (y / (z - a));
	} else if (t_1 <= 0.0004) {
		tmp = fma(-y, ((z - t) / a), x);
	} else {
		tmp = fma(((z - t) / z), y, 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+155)
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	elseif (t_1 <= 0.0004)
		tmp = fma(Float64(-y), Float64(Float64(z - t) / a), x);
	else
		tmp = fma(Float64(Float64(z - t) / z), y, 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+155], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0004], N[((-y) * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.00000000000000001e155

   1. Initial program 89.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 18.4

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 79.4

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

   8. Applied rewrites 79.4%

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

   if -1.00000000000000001e155 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000019e-4

   1. Initial program 97.9%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 91.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 89.9

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

   9. Applied rewrites 89.9%

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

   if 4.00000000000000019e-4 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. div-sub N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 87.8

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

   5. Applied rewrites 87.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-161)
     (fma (/ y a) t x)
     (if (<= t_1 2.0) (fma (/ z (- z a)) y x) (fma y (/ (- t) z) 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-161) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 2.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = fma(y, (-t / z), 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-161)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = fma(y, Float64(Float64(-t) / z), 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-161], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000003e-161

   1. Initial program 95.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if 1.00000000000000003e-161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

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

   1. Initial program 95.5%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 65.1

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 64.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-9)
     (fma y (/ t a) x)
     (if (<= t_1 2.0) (+ y x) (fma y (/ (- t) z) 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-9) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 2.0) {
		tmp = y + x;
	} else {
		tmp = fma(y, (-t / z), 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-9)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(y + x);
	else
		tmp = fma(y, Float64(Float64(-t) / z), 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-9], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y + x), $MachinePrecision], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e-9

   1. Initial program 96.0%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 74.0

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

   7. Applied rewrites 74.0%

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

   9. Applied rewrites 74.2%

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

   if 1.00000000000000006e-9 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.9

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

   5. Applied rewrites 98.9%

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

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

   1. Initial program 95.5%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 65.1

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 64.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+124}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-9)
     (fma y (/ t a) x)
     (if (<= t_1 1e+124) (+ y x) (* (- t) (/ y z))))))

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-9) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 1e+124) {
		tmp = y + x;
	} else {
		tmp = -t * (y / z);
	}
	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-9)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 1e+124)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(-t) * Float64(y / z));
	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-9], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+124], N[(y + x), $MachinePrecision], N[((-t) * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e-9

   1. Initial program 96.0%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 74.0

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

   7. Applied rewrites 74.0%

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

   9. Applied rewrites 74.2%

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

   if 1.00000000000000006e-9 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.99999999999999948e123

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if 9.99999999999999948e123 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 90.4%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 66.7

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

   7. Applied rewrites 66.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 71.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+124}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-112} \lor \neg \left(t\_1 \leq 500000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 2e-112) (not (<= t_1 500000000.0)))
     (fma (/ y a) t x)
     (+ y 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-112) || !(t_1 <= 500000000.0)) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = y + 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-112) || !(t_1 <= 500000000.0))
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(y + 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[Or[LessEqual[t$95$1, 2e-112], N[Not[LessEqual[t$95$1, 500000000.0]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-112 or 5e8 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if 1.9999999999999999e-112 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.6

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

   5. Applied rewrites 94.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-112} \lor \neg \left(\frac{z - t}{z - a} \leq 500000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 500000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-9)
     (fma y (/ t a) x)
     (if (<= t_1 500000000.0) (+ y x) (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-9) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 500000000.0) {
		tmp = y + x;
	} 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-9)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 500000000.0)
		tmp = Float64(y + x);
	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-9], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 500000000.0], N[(y + x), $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)) < 1.00000000000000006e-9

   1. Initial program 96.0%

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

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 74.0

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

   7. Applied rewrites 74.0%

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

   9. Applied rewrites 74.2%

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

   if 1.00000000000000006e-9 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 5e8 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 55.7

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

   5. Applied rewrites 55.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 500000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.6e-36)
   (fma (/ z (- z a)) y x)
   (if (<= z 1.15e-56) (fma (/ y a) t x) (fma (/ (- z t) z) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.6e-36) {
		tmp = fma((z / (z - a)), y, x);
	} else if (z <= 1.15e-56) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = fma(((z - t) / z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.6e-36)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	elseif (z <= 1.15e-56)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.6e-36], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 1.15e-56], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.59999999999999993e-36

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if -4.59999999999999993e-36 < z < 1.15000000000000001e-56

   1. Initial program 94.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if 1.15000000000000001e-56 < z

   1. Initial program 98.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. div-sub N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 88.2

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

   5. Applied rewrites 88.2%

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

Alternative 16: 98.2% 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 97.3%

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

Alternative 17: 60.0% accurate, 6.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return y + 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 = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(y + x), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 61.0

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

5. Applied rewrites 61.0%

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

Developer Target 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x + (y / ((z - a) / (z - 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 - a) / (z - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025016 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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