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

Percentage Accurate: 98.4% → 98.5%

Time: 7.9s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 - z) / (t - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f64 N/A

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

   8. neg-sub0 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. associate--r+ N/A

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

   13. neg-sub0 N/A

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

   14. remove-double-neg N/A

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

   15. lower--.f64 N/A

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

   16. neg-sub0 N/A

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

   17. lift--.f64 N/A

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

   18. sub-neg N/A

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

   19. +-commutative N/A

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

   20. associate--r+ N/A

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

   21. neg-sub0 N/A

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

   22. remove-double-neg N/A

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

   23. lower--.f64 97.2

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

4. Applied rewrites 97.2%

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

Alternative 2: 87.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y \cdot t}{a - z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* y t) (- a z))))
   (if (<= t_1 -4e+66)
     t_2
     (if (<= t_1 -0.05)
       (fma (/ (- z t) z) y x)
       (if (<= t_1 1e-19)
         (fma (- t z) (/ y a) x)
         (if (<= t_1 2e+51) (fma (/ z (- z a)) y x) t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y * t) / (a - z);
	double tmp;
	if (t_1 <= -4e+66) {
		tmp = t_2;
	} else if (t_1 <= -0.05) {
		tmp = fma(((z - t) / z), y, x);
	} else if (t_1 <= 1e-19) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 2e+51) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(y * t) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -4e+66)
		tmp = t_2;
	elseif (t_1 <= -0.05)
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	elseif (t_1 <= 1e-19)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 2e+51)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+66], t$95$2, If[LessEqual[t$95$1, -0.05], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-19], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.99999999999999978e66 or 2e51 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 92.7

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.1

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

   7. Applied rewrites 81.1%

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

   9. Applied rewrites 85.4%

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

   if -3.99999999999999978e66 < (/.f64 (-.f64 z t) (-.f64 z a)) < -0.050000000000000003

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if -0.050000000000000003 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-20

   1. Initial program 98.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 9.9999999999999998e-20 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e51

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y \cdot t}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+60}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* y t) (- a z))))
   (if (<= t_1 -1e+60)
     t_2
     (if (<= t_1 1e-19)
       (fma (- t z) (/ y a) x)
       (if (<= t_1 2e+51) (fma (/ z (- z a)) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y * t) / (a - z);
	double tmp;
	if (t_1 <= -1e+60) {
		tmp = t_2;
	} else if (t_1 <= 1e-19) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 2e+51) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(y * t) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -1e+60)
		tmp = t_2;
	elseif (t_1 <= 1e-19)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 2e+51)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+60], t$95$2, If[LessEqual[t$95$1, 1e-19], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -9.9999999999999995e59 or 2e51 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 92.8

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 80.0

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 84.2%

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

   if -9.9999999999999995e59 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-20

   1. Initial program 98.4%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if 9.9999999999999998e-20 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e51

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+60}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y \cdot t}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+60}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* y t) (- a z))))
   (if (<= t_1 -1e+60)
     t_2
     (if (<= t_1 1e-6)
       (fma (- t z) (/ y a) x)
       (if (<= t_1 2e+51) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y * t) / (a - z);
	double tmp;
	if (t_1 <= -1e+60) {
		tmp = t_2;
	} else if (t_1 <= 1e-6) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 2e+51) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(y * t) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -1e+60)
		tmp = t_2;
	elseif (t_1 <= 1e-6)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 2e+51)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+60], t$95$2, If[LessEqual[t$95$1, 1e-6], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -9.9999999999999995e59 or 2e51 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 92.8

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 80.0

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 84.2%

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

   if -9.9999999999999995e59 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.99999999999999955e-7

   1. Initial program 98.4%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if 9.99999999999999955e-7 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e51

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 93.8

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

   5. Applied rewrites 93.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+60}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{y \cdot t}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+133}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* y t) (- a z))))
   (if (<= t_1 -1e+133)
     t_2
     (if (<= t_1 1e-19) (fma y (/ t a) x) (if (<= t_1 2e+51) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (y * t) / (a - z);
	double tmp;
	if (t_1 <= -1e+133) {
		tmp = t_2;
	} else if (t_1 <= 1e-19) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 2e+51) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(y * t) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -1e+133)
		tmp = t_2;
	elseif (t_1 <= 1e-19)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 2e+51)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+133], t$95$2, If[LessEqual[t$95$1, 1e-19], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e133 or 2e51 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 80.1

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 85.1%

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

   if -1e133 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-20

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.9

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

   7. Applied rewrites 80.9%

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

   9. Applied rewrites 82.8%

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

   if 9.9999999999999998e-20 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e51

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+133}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot \frac{t}{a - z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+38}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y (/ t (- a z)))))
   (if (<= t_1 -4e+38)
     t_2
     (if (<= t_1 1e-19) (fma y (/ t a) x) (if (<= t_1 2e+51) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * (t / (a - z));
	double tmp;
	if (t_1 <= -4e+38) {
		tmp = t_2;
	} else if (t_1 <= 1e-19) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 2e+51) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(y * Float64(t / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -4e+38)
		tmp = t_2;
	elseif (t_1 <= 1e-19)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 2e+51)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+38], t$95$2, If[LessEqual[t$95$1, 1e-19], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.99999999999999991e38 or 2e51 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if -3.99999999999999991e38 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-20

   1. Initial program 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 84.6

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

   7. Applied rewrites 84.6%

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

   9. Applied rewrites 85.8%

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

   if 9.9999999999999998e-20 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e51

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{+38}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{-19} \lor \neg \left(t\_1 \leq 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -2e+151)
     (* (- t) (/ y z))
     (if (or (<= t_1 1e-19) (not (<= t_1 1e+23))) (fma y (/ t a) x) (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -2e+151) {
		tmp = -t * (y / z);
	} else if ((t_1 <= 1e-19) || !(t_1 <= 1e+23)) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -2e+151)
		tmp = Float64(Float64(-t) * Float64(y / z));
	elseif ((t_1 <= 1e-19) || !(t_1 <= 1e+23))
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+151], N[((-t) * N[(y / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 1e-19], N[Not[LessEqual[t$95$1, 1e+23]], $MachinePrecision]], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.00000000000000003e151

   1. Initial program 87.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.2%

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

   if -2.00000000000000003e151 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-20 or 9.9999999999999992e22 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 78.1

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

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 80.3%

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

   if 9.9999999999999998e-20 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999992e22

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-19} \lor \neg \left(\frac{z - t}{z - a} \leq 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-19} \lor \neg \left(t\_1 \leq 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 1e-19) (not (<= t_1 1e+23))) (fma y (/ t a) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 1e-19) || !(t_1 <= 1e+23)) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 1e-19) || !(t_1 <= 1e+23))
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 1e-19], N[Not[LessEqual[t$95$1, 1e+23]], $MachinePrecision]], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-20 or 9.9999999999999992e22 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 76.0

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

   7. Applied rewrites 76.0%

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

   9. Applied rewrites 77.9%

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

   if 9.9999999999999998e-20 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999992e22

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-19} \lor \neg \left(\frac{z - t}{z - a} \leq 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-19} \lor \neg \left(t\_1 \leq 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 1e-19) (not (<= t_1 1e+23))) (fma (/ y a) t x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 1e-19) || !(t_1 <= 1e+23)) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 1e-19) || !(t_1 <= 1e+23))
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 1e-19], N[Not[LessEqual[t$95$1, 1e+23]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-20 or 9.9999999999999992e22 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if 9.9999999999999998e-20 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999992e22

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-19} \lor \neg \left(\frac{z - t}{z - a} \leq 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+52} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+51}\right):\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 -2e+52) (not (<= t_1 2e+51))) (* y (/ t a)) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= -2e+52) || !(t_1 <= 2e+51)) {
		tmp = y * (t / a);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    if ((t_1 <= (-2d+52)) .or. (.not. (t_1 <= 2d+51))) then
        tmp = y * (t / a)
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= -2e+52) || !(t_1 <= 2e+51)) {
		tmp = y * (t / a);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if (t_1 <= -2e+52) or not (t_1 <= 2e+51):
		tmp = y * (t / a)
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= -2e+52) || !(t_1 <= 2e+51))
		tmp = Float64(y * Float64(t / a));
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	tmp = 0.0;
	if ((t_1 <= -2e+52) || ~((t_1 <= 2e+51)))
		tmp = y * (t / a);
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+52], N[Not[LessEqual[t$95$1, 2e+51]], $MachinePrecision]], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e52 or 2e51 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 80.3

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 61.9%

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

   if -2e52 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e51

   1. Initial program 99.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.6

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

   5. Applied rewrites 69.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+52} \lor \neg \left(\frac{z - t}{z - a} \leq 2 \cdot 10^{+51}\right):\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -4e+66)
     (/ (* y t) a)
     (if (<= t_1 2e+51) (+ y x) (* y (/ t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -4e+66) {
		tmp = (y * t) / a;
	} else if (t_1 <= 2e+51) {
		tmp = y + x;
	} else {
		tmp = y * (t / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    if (t_1 <= (-4d+66)) then
        tmp = (y * t) / a
    else if (t_1 <= 2d+51) then
        tmp = y + x
    else
        tmp = y * (t / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -4e+66) {
		tmp = (y * t) / a;
	} else if (t_1 <= 2e+51) {
		tmp = y + x;
	} else {
		tmp = y * (t / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if t_1 <= -4e+66:
		tmp = (y * t) / a
	elif t_1 <= 2e+51:
		tmp = y + x
	else:
		tmp = y * (t / a)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -4e+66)
		tmp = Float64(Float64(y * t) / a);
	elseif (t_1 <= 2e+51)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * Float64(t / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	tmp = 0.0;
	if (t_1 <= -4e+66)
		tmp = (y * t) / a;
	elseif (t_1 <= 2e+51)
		tmp = y + x;
	else
		tmp = y * (t / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+66], N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(y + x), $MachinePrecision], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.99999999999999978e66

   1. Initial program 92.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 70.3

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

   7. Applied rewrites 70.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.2%

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

   if -3.99999999999999978e66 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e51

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.4

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

   5. Applied rewrites 69.4%

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

   if 2e51 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 90.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 90.1

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 59.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

Alternative 13: 60.5% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(y + x), $MachinePrecision]

Derivation

1. Initial program 97.1%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 54.5

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

5. Applied rewrites 54.5%

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

Developer Target 1: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024318 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ y (/ (- z a) (- z t)))))

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