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

Percentage Accurate: 98.1% → 98.3%

Time: 8.1s

Alternatives: 14

Speedup: 1.1×

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

real(8) function code(x, y, z, t, a)
    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: 98.1% 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

real(8) function code(x, y, z, t, a)
    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: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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 / ((a - z) / (t - z))) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[x + y \cdot \frac{z - t}{z - a} \]
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. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f64 N/A

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

   8. neg-sub0 N/A

      \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

   9. lift--.f64 N/A

      \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

   10. sub-neg N/A

       \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

   11. +-commutative N/A

       \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

   12. associate--r+ N/A

       \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

   13. neg-sub0 N/A

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

   14. remove-double-neg N/A

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

   15. lower--.f64 N/A

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

   16. neg-sub0 N/A

       \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

   17. lift--.f64 N/A

       \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

   18. sub-neg N/A

       \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

   19. +-commutative N/A

       \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

   20. associate--r+ N/A

       \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

   21. neg-sub0 N/A

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

   22. remove-double-neg N/A

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

   23. lower--.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\frac{t - z}{a} \cdot y + x\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+161)
     (/ (* t y) (- a z))
     (if (<= t_1 2e-46)
       (+ (* (/ (- t z) a) y) x)
       (if (<= t_1 500000.0)
         (fma (/ z (- z a)) y x)
         (* (/ y (- z a)) (- z t)))))))

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 = (t * y) / (a - z);
	} else if (t_1 <= 2e-46) {
		tmp = (((t - z) / a) * y) + x;
	} else if (t_1 <= 500000.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = (y / (z - a)) * (z - t);
	}
	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 = Float64(Float64(t * y) / Float64(a - z));
	elseif (t_1 <= 2e-46)
		tmp = Float64(Float64(Float64(Float64(t - z) / a) * y) + x);
	elseif (t_1 <= 500000.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = Float64(Float64(y / Float64(z - a)) * Float64(z - t));
	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[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-46], N[(N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e161

   1. Initial program 95.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.0

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 90.9%

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

   if -1e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000005e-46

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. unsub-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower--.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if 2.00000000000000005e-46 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\frac{t - z}{a} \cdot y + x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+161)
     (/ (* t y) (- a z))
     (if (<= t_1 2e-46)
       (fma (- t z) (/ y a) x)
       (if (<= t_1 500000.0)
         (fma (/ z (- z a)) y x)
         (* (/ y (- z a)) (- z t)))))))

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 = (t * y) / (a - z);
	} else if (t_1 <= 2e-46) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 500000.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = (y / (z - a)) * (z - t);
	}
	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 = Float64(Float64(t * y) / Float64(a - z));
	elseif (t_1 <= 2e-46)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 500000.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = Float64(Float64(y / Float64(z - a)) * Float64(z - t));
	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[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-46], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e161

   1. Initial program 95.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.0

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 90.9%

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

   if -1e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000005e-46

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]
   4. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if 2.00000000000000005e-46 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+161)
     (/ (* t y) (- a z))
     (if (<= t_1 2e-46)
       (fma (- t z) (/ y a) x)
       (if (<= t_1 500000.0) (fma (/ z (- z a)) y x) (* (/ y (- a z)) t))))))

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 = (t * y) / (a - z);
	} else if (t_1 <= 2e-46) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 500000.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = (y / (a - z)) * t;
	}
	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 = Float64(Float64(t * y) / Float64(a - z));
	elseif (t_1 <= 2e-46)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 500000.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = Float64(Float64(y / Float64(a - z)) * t);
	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[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-46], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e161

   1. Initial program 95.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.0

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 90.9%

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

   if -1e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000005e-46

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]
   4. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if 2.00000000000000005e-46 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 5: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+161)
     (/ (* t y) (- a z))
     (if (<= t_1 1e-7)
       (fma (- t z) (/ y a) x)
       (if (<= t_1 500000.0) (+ y x) (* (/ y (- a z)) t))))))

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 = (t * y) / (a - z);
	} else if (t_1 <= 1e-7) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = (y / (a - z)) * t;
	}
	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 = Float64(Float64(t * y) / Float64(a - z));
	elseif (t_1 <= 1e-7)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 500000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / Float64(a - z)) * t);
	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[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(y + x), $MachinePrecision], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e161

   1. Initial program 95.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.0

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 90.9%

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

   if -1e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]
   4. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 6: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+161)
     (/ (* t y) (- a z))
     (if (<= t_1 1e-7)
       (+ (* (/ t a) y) x)
       (if (<= t_1 500000.0) (+ y x) (* (/ y (- a z)) t))))))

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 = (t * y) / (a - z);
	} else if (t_1 <= 1e-7) {
		tmp = ((t / a) * y) + x;
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = (y / (a - z)) * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    if (t_1 <= (-1d+161)) then
        tmp = (t * y) / (a - z)
    else if (t_1 <= 1d-7) then
        tmp = ((t / a) * y) + x
    else if (t_1 <= 500000.0d0) then
        tmp = y + x
    else
        tmp = (y / (a - z)) * 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 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -1e+161) {
		tmp = (t * y) / (a - z);
	} else if (t_1 <= 1e-7) {
		tmp = ((t / a) * y) + x;
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = (y / (a - z)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if t_1 <= -1e+161:
		tmp = (t * y) / (a - z)
	elif t_1 <= 1e-7:
		tmp = ((t / a) * y) + x
	elif t_1 <= 500000.0:
		tmp = y + x
	else:
		tmp = (y / (a - z)) * t
	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 = Float64(Float64(t * y) / Float64(a - z));
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(Float64(t / a) * y) + x);
	elseif (t_1 <= 500000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / Float64(a - z)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	tmp = 0.0;
	if (t_1 <= -1e+161)
		tmp = (t * y) / (a - z);
	elseif (t_1 <= 1e-7)
		tmp = ((t / a) * y) + x;
	elseif (t_1 <= 500000.0)
		tmp = y + x;
	else
		tmp = (y / (a - z)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+161], N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(y + x), $MachinePrecision], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e161

   1. Initial program 95.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.0

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 90.9%

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

   if -1e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.4

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

   7. Applied rewrites 81.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-7}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y}{a - z} \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+156}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* (/ y (- a z)) t)))
   (if (<= t_1 -2e+156)
     t_2
     (if (<= t_1 1e-7)
       (+ (/ (* t y) a) x)
       (if (<= t_1 500000.0) (+ 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 = (y / (a - z)) * t;
	double tmp;
	if (t_1 <= -2e+156) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = ((t * y) / a) + x;
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = (y / (a - z)) * t
    if (t_1 <= (-2d+156)) then
        tmp = t_2
    else if (t_1 <= 1d-7) then
        tmp = ((t * y) / a) + x
    else if (t_1 <= 500000.0d0) then
        tmp = y + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y / (a - z)) * t;
	double tmp;
	if (t_1 <= -2e+156) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = ((t * y) / a) + x;
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = (y / (a - z)) * t
	tmp = 0
	if t_1 <= -2e+156:
		tmp = t_2
	elif t_1 <= 1e-7:
		tmp = ((t * y) / a) + x
	elif t_1 <= 500000.0:
		tmp = 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(y / Float64(a - z)) * t)
	tmp = 0.0
	if (t_1 <= -2e+156)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(Float64(t * y) / a) + x);
	elseif (t_1 <= 500000.0)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = (y / (a - z)) * t;
	tmp = 0.0;
	if (t_1 <= -2e+156)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = ((t * y) / a) + x;
	elseif (t_1 <= 500000.0)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+156], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e156 or 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.2%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

   if -2e156 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-7}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -1e+161)
     (/ (* t y) (- a z))
     (if (<= t_1 1e-7)
       (fma (/ y a) t x)
       (if (<= t_1 500000.0) (+ y x) (* (/ y (- a z)) t))))))

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 = (t * y) / (a - z);
	} else if (t_1 <= 1e-7) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = (y / (a - z)) * t;
	}
	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 = Float64(Float64(t * y) / Float64(a - z));
	elseif (t_1 <= 1e-7)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 500000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / Float64(a - z)) * t);
	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[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(y + x), $MachinePrecision], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e161

   1. Initial program 95.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.0

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 90.9%

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

   if -1e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 83.6

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

   5. Applied rewrites 83.6%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 9: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-7)
     (fma (/ y a) t x)
     (if (<= t_1 500000.0) (+ y x) (* (/ y (- a z)) t)))))

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-7) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = (y / (a - z)) * t;
	}
	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-7)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 500000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / Float64(a - z)) * t);
	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-7], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(y + x), $MachinePrecision], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 99.2%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 81.8

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

   5. Applied rewrites 81.8%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 10: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-7)
     (fma (/ y a) t x)
     (if (<= t_1 500000.0) (+ y x) (* (/ t (- a z)) y)))))

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-7) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = (t / (a - z)) * y;
	}
	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-7)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 500000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(t / Float64(a - z)) * y);
	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-7], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(y + x), $MachinePrecision], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 99.2%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 81.8

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

   5. Applied rewrites 81.8%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y}{\frac{\color{blue}{0 - \left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z - a\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - t\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{0 - \left(z - t\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z - t\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y}{\frac{a - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.4

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 79.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+15}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ y a) t x)))
   (if (<= t_1 1e-7) t_2 (if (<= t_1 1e+15) (+ 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 = fma((y / a), t, x);
	double tmp;
	if (t_1 <= 1e-7) {
		tmp = t_2;
	} else if (t_1 <= 1e+15) {
		tmp = 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 = fma(Float64(y / a), t, x)
	tmp = 0.0
	if (t_1 <= 1e-7)
		tmp = t_2;
	elseif (t_1 <= 1e+15)
		tmp = Float64(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[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-7], t$95$2, If[LessEqual[t$95$1, 1e+15], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8 or 1e15 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.2%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 77.5

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

   5. Applied rewrites 77.5%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e15

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 96.9

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

   5. Applied rewrites 96.9%

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

Alternative 12: 65.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y}{a} \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+140}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* (/ y a) t)))
   (if (<= t_1 -2e+140) t_2 (if (<= t_1 2e+27) (+ 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 = (y / a) * t;
	double tmp;
	if (t_1 <= -2e+140) {
		tmp = t_2;
	} else if (t_1 <= 2e+27) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = (y / a) * t
    if (t_1 <= (-2d+140)) then
        tmp = t_2
    else if (t_1 <= 2d+27) then
        tmp = y + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y / a) * t;
	double tmp;
	if (t_1 <= -2e+140) {
		tmp = t_2;
	} else if (t_1 <= 2e+27) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = (y / a) * t
	tmp = 0
	if t_1 <= -2e+140:
		tmp = t_2
	elif t_1 <= 2e+27:
		tmp = 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(y / a) * t)
	tmp = 0.0
	if (t_1 <= -2e+140)
		tmp = t_2;
	elseif (t_1 <= 2e+27)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = (y / a) * t;
	tmp = 0.0;
	if (t_1 <= -2e+140)
		tmp = t_2;
	elseif (t_1 <= 2e+27)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+140], t$95$2, If[LessEqual[t$95$1, 2e+27], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.00000000000000012e140 or 2e27 < (/.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 a around inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]
   4. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 57.3%

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

   if -2.00000000000000012e140 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e27

   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 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. Add Preprocessing

Alternative 13: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 98.8

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(z - t\right)}}{\mathsf{neg}\left(\left(z - a\right)\right)}, y, x\right) \]

   10. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(z - t\right)}}{\mathsf{neg}\left(\left(z - a\right)\right)}, y, x\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}{\mathsf{neg}\left(\left(z - a\right)\right)}, y, x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}}{\mathsf{neg}\left(\left(z - a\right)\right)}, y, x\right) \]

   13. associate--r+ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}}{\mathsf{neg}\left(\left(z - a\right)\right)}, y, x\right) \]

   14. neg-sub0 N/A

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

   15. remove-double-neg N/A

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

   16. lower--.f64 N/A

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

   17. neg-sub0 N/A

       \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{0 - \left(z - a\right)}}, y, x\right) \]

   18. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{t - z}{0 - \color{blue}{\left(z - a\right)}}, y, x\right) \]

   19. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{t - z}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(a\right)\right)\right)}}, y, x\right) \]

   20. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{t - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) + z\right)}}, y, x\right) \]

   21. associate--r+ N/A

       \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a\right)\right)\right) - z}}, y, x\right) \]

   22. neg-sub0 N/A

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

   23. remove-double-neg N/A

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

   24. lower--.f64 98.8

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

4. Applied rewrites 98.8%

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

Alternative 14: 61.1% 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

real(8) function code(x, y, z, t, a)
    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 98.8%

   \[x + y \cdot \frac{z - t}{z - a} \]
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 57.3

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

5. Applied rewrites 57.3%

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

Developer Target 1: 98.3% 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

real(8) function code(x, y, z, t, a)
    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 2024248 
(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)))))