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

Percentage Accurate: 97.8% → 98.0%

Time: 10.1s

Alternatives: 12

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: 97.8% 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: 98.0% 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]

Derivation

1. Initial program 97.7%

   \[x + y \cdot \frac{z - t}{z - a} \]
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. lower-/.f64 98.1

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

4. Applied rewrites 98.1%

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

Alternative 2: 80.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;t\_1 \leq 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{-a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+18)
     (* y (/ t (- a z)))
     (if (<= t_1 1e-48)
       (fma y (/ z (- a)) x)
       (if (<= t_1 1e+17)
         (+ x y)
         (if (<= t_1 2e+150) (fma y (/ (- t) z) x) (fma t (/ y a) 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 <= -5e+18) {
		tmp = y * (t / (a - z));
	} else if (t_1 <= 1e-48) {
		tmp = fma(y, (z / -a), x);
	} else if (t_1 <= 1e+17) {
		tmp = x + y;
	} else if (t_1 <= 2e+150) {
		tmp = fma(y, (-t / z), x);
	} else {
		tmp = fma(t, (y / a), 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 <= -5e+18)
		tmp = Float64(y * Float64(t / Float64(a - z)));
	elseif (t_1 <= 1e-48)
		tmp = fma(y, Float64(z / Float64(-a)), x);
	elseif (t_1 <= 1e+17)
		tmp = Float64(x + y);
	elseif (t_1 <= 2e+150)
		tmp = fma(y, Float64(Float64(-t) / z), x);
	else
		tmp = fma(t, Float64(y / a), 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, -5e+18], N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-48], N[(y * N[(z / (-a)), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+17], N[(x + y), $MachinePrecision], If[LessEqual[t$95$1, 2e+150], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5e18

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 77.1

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

   7. Applied rewrites 77.1%

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

   if -5e18 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999997e-49

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{z - a}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 86.1%

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

   if 9.9999999999999997e-49 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e17

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if 1e17 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999996e150

   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. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 82.0%

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

   if 1.99999999999999996e150 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 68.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{-a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;t\_1 \leq 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(z, -\frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+18)
     (* y (/ t (- a z)))
     (if (<= t_1 1e-48)
       (fma z (- (/ y a)) x)
       (if (<= t_1 1e+17)
         (+ x y)
         (if (<= t_1 2e+150) (fma y (/ (- t) z) x) (fma t (/ y a) 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 <= -5e+18) {
		tmp = y * (t / (a - z));
	} else if (t_1 <= 1e-48) {
		tmp = fma(z, -(y / a), x);
	} else if (t_1 <= 1e+17) {
		tmp = x + y;
	} else if (t_1 <= 2e+150) {
		tmp = fma(y, (-t / z), x);
	} else {
		tmp = fma(t, (y / a), 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 <= -5e+18)
		tmp = Float64(y * Float64(t / Float64(a - z)));
	elseif (t_1 <= 1e-48)
		tmp = fma(z, Float64(-Float64(y / a)), x);
	elseif (t_1 <= 1e+17)
		tmp = Float64(x + y);
	elseif (t_1 <= 2e+150)
		tmp = fma(y, Float64(Float64(-t) / z), x);
	else
		tmp = fma(t, Float64(y / a), 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, -5e+18], N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-48], N[(z * (-N[(y / a), $MachinePrecision]) + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+17], N[(x + y), $MachinePrecision], If[LessEqual[t$95$1, 2e+150], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5e18

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 77.1

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

   7. Applied rewrites 77.1%

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

   if -5e18 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999997e-49

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{z - a}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 82.4%

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

   if 9.9999999999999997e-49 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e17

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if 1e17 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999996e150

   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. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 82.0%

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

   if 1.99999999999999996e150 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 68.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(z, -\frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{-a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+18)
     (* y (/ t (- a z)))
     (if (<= t_1 0.5)
       (fma y (/ z (- a)) x)
       (if (<= t_1 2e+150) (fma y (- 1.0 (/ t z)) x) (fma t (/ y a) 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 <= -5e+18) {
		tmp = y * (t / (a - z));
	} else if (t_1 <= 0.5) {
		tmp = fma(y, (z / -a), x);
	} else if (t_1 <= 2e+150) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else {
		tmp = fma(t, (y / a), 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 <= -5e+18)
		tmp = Float64(y * Float64(t / Float64(a - z)));
	elseif (t_1 <= 0.5)
		tmp = fma(y, Float64(z / Float64(-a)), x);
	elseif (t_1 <= 2e+150)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	else
		tmp = fma(t, Float64(y / a), 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, -5e+18], N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], N[(y * N[(z / (-a)), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+150], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5e18

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 77.1

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

   7. Applied rewrites 77.1%

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

   if -5e18 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.5

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{z - a}} \]
   6. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 88.6

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

   7. Applied rewrites 88.6%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 85.0%

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

   if 0.5 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999996e150

   1. Initial program 100.0%

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

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if 1.99999999999999996e150 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 68.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

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

Alternative 5: 82.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;t\_1 \leq 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+18)
     (* y (/ t (- a z)))
     (if (<= t_1 1e+17)
       (fma y (/ z (- z a)) x)
       (if (<= t_1 2e+150) (fma y (/ (- t) z) x) (fma t (/ y a) 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 <= -5e+18) {
		tmp = y * (t / (a - z));
	} else if (t_1 <= 1e+17) {
		tmp = fma(y, (z / (z - a)), x);
	} else if (t_1 <= 2e+150) {
		tmp = fma(y, (-t / z), x);
	} else {
		tmp = fma(t, (y / a), 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 <= -5e+18)
		tmp = Float64(y * Float64(t / Float64(a - z)));
	elseif (t_1 <= 1e+17)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	elseif (t_1 <= 2e+150)
		tmp = fma(y, Float64(Float64(-t) / z), x);
	else
		tmp = fma(t, Float64(y / a), 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, -5e+18], N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+17], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+150], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5e18

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 77.1

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

   7. Applied rewrites 77.1%

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

   if -5e18 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e17

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 93.6

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

   5. Applied rewrites 93.6%

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

   if 1e17 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999996e150

   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. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 82.0%

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

   if 1.99999999999999996e150 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 68.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 0.01)
     (fma y (/ t a) x)
     (if (<= t_1 1e+17)
       (+ x y)
       (if (<= t_1 2e+150) (fma y (/ (- t) z) x) (fma t (/ y a) 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 <= 0.01) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 1e+17) {
		tmp = x + y;
	} else if (t_1 <= 2e+150) {
		tmp = fma(y, (-t / z), x);
	} else {
		tmp = fma(t, (y / a), 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 <= 0.01)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 1e+17)
		tmp = Float64(x + y);
	elseif (t_1 <= 2e+150)
		tmp = fma(y, Float64(Float64(-t) / z), x);
	else
		tmp = fma(t, Float64(y / a), 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, 0.01], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+17], N[(x + y), $MachinePrecision], If[LessEqual[t$95$1, 2e+150], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0100000000000000002

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 0.0100000000000000002 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e17

   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 97.1

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

   5. Applied rewrites 97.1%

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

   if 1e17 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999996e150

   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. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 82.0%

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

   if 1.99999999999999996e150 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 68.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+17}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+18)
     (* y (/ t (- a z)))
     (if (<= t_1 2e+17) (fma y (/ z (- z a)) x) (/ (* y t) (- a z))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+18) {
		tmp = y * (t / (a - z));
	} else if (t_1 <= 2e+17) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = (y * t) / (a - z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+18)
		tmp = Float64(y * Float64(t / Float64(a - z)));
	elseif (t_1 <= 2e+17)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = Float64(Float64(y * t) / Float64(a - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+18], N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+17], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5e18

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 77.1

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

   7. Applied rewrites 77.1%

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

   if -5e18 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e17

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 93.7

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

   5. Applied rewrites 93.7%

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

   if 2e17 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 86.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. neg-mul-1 N/A

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

      11. lower-+.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 76.9

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

   5. Applied rewrites 76.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 0.01)
     (fma y (/ t a) x)
     (if (<= t_1 500.0) (+ x y) (fma t (/ y a) 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 <= 0.01) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 500.0) {
		tmp = x + y;
	} else {
		tmp = fma(t, (y / a), 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 <= 0.01)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 500.0)
		tmp = Float64(x + y);
	else
		tmp = fma(t, Float64(y / a), 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, 0.01], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 500.0], N[(x + y), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0100000000000000002

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 0.0100000000000000002 < (/.f64 (-.f64 z t) (-.f64 z a)) < 500

   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 98.0

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

   5. Applied rewrites 98.0%

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

   if 500 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 87.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 91.5

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 61.2

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

   7. Applied rewrites 61.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 500:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma y (/ t a) x)))
   (if (<= t_1 0.01) t_2 (if (<= t_1 500.0) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(y, (t / a), x);
	double tmp;
	if (t_1 <= 0.01) {
		tmp = t_2;
	} else if (t_1 <= 500.0) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(y, Float64(t / a), x)
	tmp = 0.0
	if (t_1 <= 0.01)
		tmp = t_2;
	elseif (t_1 <= 500.0)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 0.01], t$95$2, If[LessEqual[t$95$1, 500.0], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 0.0100000000000000002 or 500 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.3%

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if 0.0100000000000000002 < (/.f64 (-.f64 z t) (-.f64 z a)) < 500

   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 500:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.8% 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 97.7%

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

Alternative 11: 96.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 93.6

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

4. Applied rewrites 93.6%

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

Alternative 12: 60.3% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 61.1

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

5. Applied rewrites 61.1%

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

6. Final simplification 61.1%

   \[\leadsto x + y \]
7. Add Preprocessing

Developer Target 1: 98.0% 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 2024222 
(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)))))