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

Percentage Accurate: 98.2% → 98.4%

Time: 10.2s

Alternatives: 16

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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) / (a - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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) / (a - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. --lowering--.f64 99.2

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

4. Applied egg-rr 99.2%

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

Alternative 2: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -4e+37)
     (* z (/ y (- a t)))
     (if (<= t_1 2e-13)
       (fma y (/ (- z t) a) x)
       (if (<= t_1 2.0) (fma (/ t (- t a)) y x) (* y (/ z (- a t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -4e+37) {
		tmp = z * (y / (a - t));
	} else if (t_1 <= 2e-13) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 2.0) {
		tmp = fma((t / (t - a)), y, x);
	} else {
		tmp = y * (z / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -4e+37)
		tmp = Float64(z * Float64(y / Float64(a - t)));
	elseif (t_1 <= 2e-13)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(t / Float64(t - a)), y, x);
	else
		tmp = Float64(y * Float64(z / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+37], N[(z * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-13], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -3.99999999999999982e37

   1. Initial program 94.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. neg-sub0 N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. neg-sub0 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 85.5

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

   7. Simplified 85.5%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. distribute-neg-in N/A

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

      8. sub-neg N/A

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

      9. frac-2neg N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 85.5

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

   9. Applied egg-rr 85.5%

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

   if -3.99999999999999982e37 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2.0000000000000001e-13

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 97.5

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

   5. Simplified 97.5%

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

   if 2.0000000000000001e-13 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 98.8

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

   7. Simplified 98.8%

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

      1. remove-double-neg N/A

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

      2. frac-2neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. neg-sub0 N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 98.8

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

   9. Applied egg-rr 98.8%

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

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

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 66.5

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

   5. Simplified 66.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 40000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -4e+37)
     (* z (/ y (- a t)))
     (if (<= t_1 0.2)
       (fma y (/ (- z t) a) x)
       (if (<= t_1 40000000.0)
         (fma y (- 1.0 (/ z t)) x)
         (* y (/ z (- a t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -4e+37) {
		tmp = z * (y / (a - t));
	} else if (t_1 <= 0.2) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 40000000.0) {
		tmp = fma(y, (1.0 - (z / t)), x);
	} else {
		tmp = y * (z / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -4e+37)
		tmp = Float64(z * Float64(y / Float64(a - t)));
	elseif (t_1 <= 0.2)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 40000000.0)
		tmp = fma(y, Float64(1.0 - Float64(z / t)), x);
	else
		tmp = Float64(y * Float64(z / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+37], N[(z * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.2], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 40000000.0], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -3.99999999999999982e37

   1. Initial program 94.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. neg-sub0 N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. neg-sub0 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 85.5

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

   7. Simplified 85.5%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. distribute-neg-in N/A

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

      8. sub-neg N/A

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

      9. frac-2neg N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 85.5

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

   9. Applied egg-rr 85.5%

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

   if -3.99999999999999982e37 < (/.f64 (-.f64 z t) (-.f64 a t)) < 0.20000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 95.6

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

   5. Simplified 95.6%

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

   if 0.20000000000000001 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4e7

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. /-lowering-/.f64 99.8

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

   5. Simplified 99.8%

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

   if 4e7 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 66.8

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

   5. Simplified 66.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 40000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 40000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -4e+37)
     (* z (/ y (- a t)))
     (if (<= t_1 5e-11)
       (fma y (/ z a) x)
       (if (<= t_1 40000000.0)
         (fma y (- 1.0 (/ z t)) x)
         (* y (/ z (- a t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -4e+37) {
		tmp = z * (y / (a - t));
	} else if (t_1 <= 5e-11) {
		tmp = fma(y, (z / a), x);
	} else if (t_1 <= 40000000.0) {
		tmp = fma(y, (1.0 - (z / t)), x);
	} else {
		tmp = y * (z / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -4e+37)
		tmp = Float64(z * Float64(y / Float64(a - t)));
	elseif (t_1 <= 5e-11)
		tmp = fma(y, Float64(z / a), x);
	elseif (t_1 <= 40000000.0)
		tmp = fma(y, Float64(1.0 - Float64(z / t)), x);
	else
		tmp = Float64(y * Float64(z / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+37], N[(z * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-11], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 40000000.0], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -3.99999999999999982e37

   1. Initial program 94.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. neg-sub0 N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. neg-sub0 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 85.5

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

   7. Simplified 85.5%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. distribute-neg-in N/A

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

      8. sub-neg N/A

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

      9. frac-2neg N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 85.5

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

   9. Applied egg-rr 85.5%

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

   if -3.99999999999999982e37 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000018e-11

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 86.9

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

   5. Simplified 86.9%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4e7

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. /-lowering-/.f64 98.2

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

   5. Simplified 98.2%

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

   if 4e7 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 66.8

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

   5. Simplified 66.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 40000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -4e+37)
     (* z (/ y (- a t)))
     (if (<= t_1 5e-11)
       (fma y (/ z a) x)
       (if (<= t_1 2.0) (+ x y) (* y (/ z (- a t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -4e+37) {
		tmp = z * (y / (a - t));
	} else if (t_1 <= 5e-11) {
		tmp = fma(y, (z / a), x);
	} else if (t_1 <= 2.0) {
		tmp = x + y;
	} else {
		tmp = y * (z / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -4e+37)
		tmp = Float64(z * Float64(y / Float64(a - t)));
	elseif (t_1 <= 5e-11)
		tmp = fma(y, Float64(z / a), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(z / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+37], N[(z * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-11], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x + y), $MachinePrecision], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -3.99999999999999982e37

   1. Initial program 94.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. neg-sub0 N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. neg-sub0 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 85.5

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

   7. Simplified 85.5%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. distribute-neg-in N/A

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

      8. sub-neg N/A

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

      9. frac-2neg N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 85.5

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

   9. Applied egg-rr 85.5%

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

   if -3.99999999999999982e37 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000018e-11

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 86.9

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

   5. Simplified 86.9%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 97.1

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

   5. Simplified 97.1%

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

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

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 66.5

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

   5. Simplified 66.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := y \cdot \frac{z}{a - t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+37}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (* y (/ z (- a t)))))
   (if (<= t_1 -4e+37)
     t_2
     (if (<= t_1 5e-11) (fma y (/ z a) x) (if (<= t_1 2.0) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = y * (z / (a - t));
	double tmp;
	if (t_1 <= -4e+37) {
		tmp = t_2;
	} else if (t_1 <= 5e-11) {
		tmp = fma(y, (z / a), x);
	} else if (t_1 <= 2.0) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(y * Float64(z / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -4e+37)
		tmp = t_2;
	elseif (t_1 <= 5e-11)
		tmp = fma(y, Float64(z / a), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+37], t$95$2, If[LessEqual[t$95$1, 5e-11], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -3.99999999999999982e37 or 2 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 72.8

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

   5. Simplified 72.8%

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

   if -3.99999999999999982e37 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000018e-11

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 86.9

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

   5. Simplified 86.9%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 97.1

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

   5. Simplified 97.1%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -4 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ t_2 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \left(-\frac{y}{t}\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ z a) x)) (t_2 (/ (- z t) (- a t))))
   (if (<= t_2 -4e+37)
     (* z (- (/ y t)))
     (if (<= t_2 5e-11) t_1 (if (<= t_2 1e+19) (+ x y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (z / a), x);
	double t_2 = (z - t) / (a - t);
	double tmp;
	if (t_2 <= -4e+37) {
		tmp = z * -(y / t);
	} else if (t_2 <= 5e-11) {
		tmp = t_1;
	} else if (t_2 <= 1e+19) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(z / a), x)
	t_2 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_2 <= -4e+37)
		tmp = Float64(z * Float64(-Float64(y / t)));
	elseif (t_2 <= 5e-11)
		tmp = t_1;
	elseif (t_2 <= 1e+19)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+37], N[(z * (-N[(y / t), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, 5e-11], t$95$1, If[LessEqual[t$95$2, 1e+19], N[(x + y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -3.99999999999999982e37

   1. Initial program 94.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 80.2

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

   5. Simplified 80.2%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-lowering-neg.f64 51.0

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

   8. Simplified 51.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. /-lowering-/.f64 59.4

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

   10. Applied egg-rr 59.4%

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

   if -3.99999999999999982e37 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000018e-11 or 1e19 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 79.0

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

   5. Simplified 79.0%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1e19

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 95.5

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

   5. Simplified 95.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -4 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \left(-\frac{y}{t}\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;t\_1 \leq 6 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 1.000000000000001:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{x}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -1e+18)
     (* z (/ y a))
     (if (<= t_1 6e-104)
       x
       (if (<= t_1 1.000000000000001) (+ x y) (fma x (/ y x) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = z * (y / a);
	} else if (t_1 <= 6e-104) {
		tmp = x;
	} else if (t_1 <= 1.000000000000001) {
		tmp = x + y;
	} else {
		tmp = fma(x, (y / x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = Float64(z * Float64(y / a));
	elseif (t_1 <= 6e-104)
		tmp = x;
	elseif (t_1 <= 1.000000000000001)
		tmp = Float64(x + y);
	else
		tmp = fma(x, Float64(y / x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+18], N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 6e-104], x, If[LessEqual[t$95$1, 1.000000000000001], N[(x + y), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e18

   1. Initial program 95.1%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. neg-sub0 N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. neg-sub0 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 84.8

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

   7. Simplified 84.8%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 43.4

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

   10. Simplified 43.4%

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

   if -1e18 < (/.f64 (-.f64 z t) (-.f64 a t)) < 6.0000000000000005e-104

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 76.4%

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

   if 6.0000000000000005e-104 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.00000000000000111

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 93.1

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

   5. Simplified 93.1%

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

   if 1.00000000000000111 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.5%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 38.9

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

   5. Simplified 38.9%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{x} + x \cdot 1} \]

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 50.1

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

   8. Simplified 50.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 6 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 1.000000000000001:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{x}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma y (/ z a) x)))
   (if (<= t_1 5e-11) t_2 (if (<= t_1 1e+19) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma(y, (z / a), x);
	double tmp;
	if (t_1 <= 5e-11) {
		tmp = t_2;
	} else if (t_1 <= 1e+19) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(y, Float64(z / a), x)
	tmp = 0.0
	if (t_1 <= 5e-11)
		tmp = t_2;
	elseif (t_1 <= 1e+19)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-11], t$95$2, If[LessEqual[t$95$1, 1e+19], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000018e-11 or 1e19 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 98.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 71.9

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

   5. Simplified 71.9%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1e19

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 95.5

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

   5. Simplified 95.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+121}:\\ \;\;\;\;y\\ \mathbf{elif}\;t\_1 \leq 10^{+59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= t_1 -5e+121) y (if (<= t_1 1e+59) x y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t_1 <= -5e+121) {
		tmp = y;
	} else if (t_1 <= 1e+59) {
		tmp = x;
	} else {
		tmp = y;
	}
	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 = y * ((z - t) / (a - t))
    if (t_1 <= (-5d+121)) then
        tmp = y
    else if (t_1 <= 1d+59) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t_1 <= -5e+121) {
		tmp = y;
	} else if (t_1 <= 1e+59) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (a - t))
	tmp = 0
	if t_1 <= -5e+121:
		tmp = y
	elif t_1 <= 1e+59:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -5e+121)
		tmp = y;
	elseif (t_1 <= 1e+59)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (a - t));
	tmp = 0.0;
	if (t_1 <= -5e+121)
		tmp = y;
	elseif (t_1 <= 1e+59)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+121], y, If[LessEqual[t$95$1, 1e+59], x, y]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < -5.00000000000000007e121 or 9.99999999999999972e58 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))

   1. Initial program 97.1%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 45.6

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

   5. Simplified 45.6%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 36.2%

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

   if -5.00000000000000007e121 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < 9.99999999999999972e58

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 70.1%

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

Alternative 11: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;t\_1 \leq 6 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -1e+18) (* z (/ y a)) (if (<= t_1 6e-104) x (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = z * (y / a);
	} else if (t_1 <= 6e-104) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	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) / (a - t)
    if (t_1 <= (-1d+18)) then
        tmp = z * (y / a)
    else if (t_1 <= 6d-104) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = z * (y / a);
	} else if (t_1 <= 6e-104) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	tmp = 0
	if t_1 <= -1e+18:
		tmp = z * (y / a)
	elif t_1 <= 6e-104:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = Float64(z * Float64(y / a));
	elseif (t_1 <= 6e-104)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	tmp = 0.0;
	if (t_1 <= -1e+18)
		tmp = z * (y / a);
	elseif (t_1 <= 6e-104)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+18], N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 6e-104], x, N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e18

   1. Initial program 95.1%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. neg-sub0 N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. neg-sub0 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 84.8

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

   7. Simplified 84.8%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 43.4

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

   10. Simplified 43.4%

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

   if -1e18 < (/.f64 (-.f64 z t) (-.f64 a t)) < 6.0000000000000005e-104

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 76.4%

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

   if 6.0000000000000005e-104 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.3%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 78.6

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

   5. Simplified 78.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 6 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t\_1 \leq 6 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -1e+18) (* y (/ z a)) (if (<= t_1 6e-104) x (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = y * (z / a);
	} else if (t_1 <= 6e-104) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	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) / (a - t)
    if (t_1 <= (-1d+18)) then
        tmp = y * (z / a)
    else if (t_1 <= 6d-104) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = y * (z / a);
	} else if (t_1 <= 6e-104) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	tmp = 0
	if t_1 <= -1e+18:
		tmp = y * (z / a)
	elif t_1 <= 6e-104:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = Float64(y * Float64(z / a));
	elseif (t_1 <= 6e-104)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	tmp = 0.0;
	if (t_1 <= -1e+18)
		tmp = y * (z / a);
	elseif (t_1 <= 6e-104)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+18], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 6e-104], x, N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e18

   1. Initial program 95.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 80.1

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

   5. Simplified 80.1%

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

   6. Taylor expanded in a around inf

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

   8. Simplified 43.3%

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

   if -1e18 < (/.f64 (-.f64 z t) (-.f64 a t)) < 6.0000000000000005e-104

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 76.4%

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

   if 6.0000000000000005e-104 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.3%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 78.6

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

   5. Simplified 78.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 6 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 9 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- a t)) 9e-104) x (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= 9e-104) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	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) :: tmp
    if (((z - t) / (a - t)) <= 9d-104) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= 9e-104) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (a - t)) <= 9e-104:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(a - t)) <= 9e-104)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (a - t)) <= 9e-104)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], 9e-104], x, N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 8.9999999999999995e-104

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 52.2%

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

   if 8.9999999999999995e-104 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.3%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 78.6

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

   5. Simplified 78.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 9 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. --lowering--.f64 98.8

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

4. Applied egg-rr 98.8%

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

Alternative 15: 96.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. +-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. frac-2neg N/A

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

   5. associate-/r/ N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. neg-sub0 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. associate--r+ N/A

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

   12. neg-sub0 N/A

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

   13. remove-double-neg N/A

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

   14. --lowering--.f64 N/A

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

   15. neg-sub0 N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. associate--r+ N/A

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

   19. neg-sub0 N/A

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

   20. remove-double-neg N/A

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

   21. --lowering--.f64 96.9

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

4. Applied egg-rr 96.9%

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

Alternative 16: 51.1% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x around inf

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

5. Simplified 46.9%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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 = x + (y * ((z - t) / (a - t)))
    if (y < (-8.508084860551241d-17)) then
        tmp = t_1
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) * (1.0d0 / (a - t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

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

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