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

Percentage Accurate: 98.4% → 98.4%

Time: 8.0s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% 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: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 2: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{-a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{\left(-z\right) + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -2e+43)
     (fma y (/ (- t) z) x)
     (if (<= t_1 2e-19)
       (fma (/ (- z t) (- a)) y x)
       (if (<= t_1 1e+64) (fma (/ z (- z a)) y x) (/ (* y t) (+ (- z) 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 <= -2e+43) {
		tmp = fma(y, (-t / z), x);
	} else if (t_1 <= 2e-19) {
		tmp = fma(((z - t) / -a), y, x);
	} else if (t_1 <= 1e+64) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = (y * t) / (-z + 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 <= -2e+43)
		tmp = fma(y, Float64(Float64(-t) / z), x);
	elseif (t_1 <= 2e-19)
		tmp = fma(Float64(Float64(z - t) / Float64(-a)), y, x);
	elseif (t_1 <= 1e+64)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = Float64(Float64(y * t) / Float64(Float64(-z) + a));
	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+43], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-19], N[(N[(N[(z - t), $MachinePrecision] / (-a)), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+64], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y * t), $MachinePrecision] / N[((-z) + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.00000000000000003e43

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 84.6%

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

   if -2.00000000000000003e43 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e-19

   1. Initial program 98.7%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. lower-fma.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.4

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

   7. Applied rewrites 95.4%

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

   if 2e-19 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000002e64

   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 92.3

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

   5. Applied rewrites 92.3%

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

   if 1.00000000000000002e64 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 85.5%

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 74.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{-a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{\left(-z\right) + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{\left(-z\right) + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -2e+43)
     (fma y (/ (- t) z) x)
     (if (<= t_1 1e-52)
       (fma (/ t a) y x)
       (if (<= t_1 1e+64) (fma (/ z (- z a)) y x) (/ (* y t) (+ (- z) 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 <= -2e+43) {
		tmp = fma(y, (-t / z), x);
	} else if (t_1 <= 1e-52) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 1e+64) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = (y * t) / (-z + 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 <= -2e+43)
		tmp = fma(y, Float64(Float64(-t) / z), x);
	elseif (t_1 <= 1e-52)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 1e+64)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = Float64(Float64(y * t) / Float64(Float64(-z) + a));
	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+43], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-52], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+64], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y * t), $MachinePrecision] / N[((-z) + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.00000000000000003e43

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 84.6%

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

   if -2.00000000000000003e43 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-52

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.9

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

   7. Applied rewrites 85.9%

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

   if 1e-52 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000002e64

   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 91.7

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

   5. Applied rewrites 91.7%

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

   if 1.00000000000000002e64 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 85.5%

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 74.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{\left(-z\right) + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -2e+43)
     (fma y (/ (- t) z) x)
     (if (<= t_1 1e-52)
       (fma (/ t a) y x)
       (if (<= t_1 2.0) (fma (/ z (- z a)) y x) (+ x (/ (* t y) 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 <= -2e+43) {
		tmp = fma(y, (-t / z), x);
	} else if (t_1 <= 1e-52) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = x + ((t * y) / 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 <= -2e+43)
		tmp = fma(y, Float64(Float64(-t) / z), x);
	elseif (t_1 <= 1e-52)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = Float64(x + Float64(Float64(t * y) / a));
	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+43], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-52], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.00000000000000003e43

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 84.6%

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

   if -2.00000000000000003e43 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-52

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.9

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

   7. Applied rewrites 85.9%

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

   if 1e-52 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   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 97.2

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

   5. Applied rewrites 97.2%

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

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

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

Alternative 5: 81.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -2e+43)
     (fma y (/ (- t) z) x)
     (if (<= t_1 2e-19)
       (fma (/ t a) y x)
       (if (<= t_1 2e+31) (+ y x) (fma (/ y a) t x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -2e+43) {
		tmp = fma(y, (-t / z), x);
	} else if (t_1 <= 2e-19) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2e+31) {
		tmp = y + x;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -2e+43)
		tmp = fma(y, Float64(Float64(-t) / z), x);
	elseif (t_1 <= 2e-19)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2e+31)
		tmp = Float64(y + x);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+43], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-19], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+31], N[(y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.00000000000000003e43

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 84.6%

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

   if -2.00000000000000003e43 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e-19

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if 2e-19 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e31

   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%

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

   if 1.9999999999999999e31 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 89.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 66.2

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

   5. Applied rewrites 66.2%

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

Alternative 6: 79.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+74}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -4e+74)
     (* (- t) (/ y z))
     (if (<= t_1 2e-19)
       (fma (/ t a) y x)
       (if (<= t_1 2e+31) (+ y x) (fma (/ y a) t x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -4e+74) {
		tmp = -t * (y / z);
	} else if (t_1 <= 2e-19) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2e+31) {
		tmp = y + x;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -4e+74)
		tmp = Float64(Float64(-t) * Float64(y / z));
	elseif (t_1 <= 2e-19)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2e+31)
		tmp = Float64(y + x);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+74], N[((-t) * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-19], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+31], N[(y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.99999999999999981e74

   1. Initial program 99.6%

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 60.4%

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

   if -3.99999999999999981e74 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e-19

   1. Initial program 98.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 81.5

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

   5. Applied rewrites 81.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.5

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

   7. Applied rewrites 81.5%

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

   if 2e-19 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e31

   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%

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

   if 1.9999999999999999e31 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 89.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 66.2

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

   5. Applied rewrites 66.2%

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

Alternative 7: 81.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-19} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+31}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 2e-19) (not (<= t_1 2e+31))) (fma (/ y a) t x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 2e-19) || !(t_1 <= 2e+31)) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 2e-19) || !(t_1 <= 2e+31))
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 2e-19], N[Not[LessEqual[t$95$1, 2e+31]], $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)) < 2e-19 or 1.9999999999999999e31 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.7%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   if 2e-19 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e31

   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 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-19} \lor \neg \left(\frac{z - t}{z - a} \leq 2 \cdot 10^{+31}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-19)
     (fma (/ t a) y x)
     (if (<= t_1 2e+31) (+ y x) (fma (/ y a) t x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-19) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2e+31) {
		tmp = y + x;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-19)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2e+31)
		tmp = Float64(y + x);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-19], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+31], N[(y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2e-19

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 72.3

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

   5. Applied rewrites 72.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 72.3

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

   7. Applied rewrites 72.3%

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

   if 2e-19 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e31

   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%

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

   if 1.9999999999999999e31 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 89.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 66.2

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

   5. Applied rewrites 66.2%

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

Alternative 9: 64.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+135} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 -4e+135) (not (<= t_1 2e+101))) (/ (* t y) 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 <= -4e+135) || !(t_1 <= 2e+101)) {
		tmp = (t * y) / 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 <= (-4d+135)) .or. (.not. (t_1 <= 2d+101))) then
        tmp = (t * y) / 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 <= -4e+135) || !(t_1 <= 2e+101)) {
		tmp = (t * y) / 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 <= -4e+135) or not (t_1 <= 2e+101):
		tmp = (t * y) / 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 <= -4e+135) || !(t_1 <= 2e+101))
		tmp = Float64(Float64(t * y) / 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 <= -4e+135) || ~((t_1 <= 2e+101)))
		tmp = (t * y) / 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, -4e+135], N[Not[LessEqual[t$95$1, 2e+101]], $MachinePrecision]], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.99999999999999985e135 or 2e101 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 89.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.0%

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

   if -3.99999999999999985e135 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e101

   1. Initial program 99.4%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 68.4

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

   5. Applied rewrites 68.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{+135} \lor \neg \left(\frac{z - t}{z - a} \leq 2 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+135} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+101}\right):\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 -4e+135) (not (<= t_1 2e+101))) (* 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 <= -4e+135) || !(t_1 <= 2e+101)) {
		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 <= (-4d+135)) .or. (.not. (t_1 <= 2d+101))) 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 <= -4e+135) || !(t_1 <= 2e+101)) {
		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 <= -4e+135) or not (t_1 <= 2e+101):
		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 <= -4e+135) || !(t_1 <= 2e+101))
		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 <= -4e+135) || ~((t_1 <= 2e+101)))
		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, -4e+135], N[Not[LessEqual[t$95$1, 2e+101]], $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)) < -3.99999999999999985e135 or 2e101 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 89.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.0%

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

   10. Applied rewrites 49.3%

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

   if -3.99999999999999985e135 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e101

   1. Initial program 99.4%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 68.4

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

   5. Applied rewrites 68.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{+135} \lor \neg \left(\frac{z - t}{z - a} \leq 2 \cdot 10^{+101}\right):\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.5% 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 98.0%

   \[x + y \cdot \frac{z - t}{z - a} \]
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 61.0

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

5. Applied rewrites 61.0%

   \[\leadsto \color{blue}{y + x} \]
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 2024338 
(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)))))