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

Percentage Accurate: 98.1% → 98.3%

Time: 9.5s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

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

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 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.0

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

4. Applied rewrites 99.0%

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

Alternative 2: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;t\_1 \leq -1400000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -2e+72)
     (* t (/ y (- a z)))
     (if (<= t_1 -1400000000.0)
       (fma y (- 1.0 (/ t z)) x)
       (if (<= t_1 2e-22)
         (fma y (/ (- t z) a) x)
         (if (<= t_1 10000000.0) (+ x y) (fma y (/ t 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 <= -2e+72) {
		tmp = t * (y / (a - z));
	} else if (t_1 <= -1400000000.0) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else if (t_1 <= 2e-22) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 10000000.0) {
		tmp = x + y;
	} else {
		tmp = fma(y, (t / 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 <= -2e+72)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	elseif (t_1 <= -1400000000.0)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	elseif (t_1 <= 2e-22)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 10000000.0)
		tmp = Float64(x + y);
	else
		tmp = fma(y, Float64(t / 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, -2e+72], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1400000000.0], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-22], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 10000000.0], N[(x + y), $MachinePrecision], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.99999999999999989e72

   1. Initial program 96.3%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 96.3

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

   4. Applied rewrites 96.3%

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

   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. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 z a)) < -1.4e9

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. 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 85.3

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

   5. Applied rewrites 85.3%

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

   if -1.4e9 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 2.0000000000000001e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e7

   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 98.4

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

   5. Applied rewrites 98.4%

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

   if 1e7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-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 74.5

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

   5. Applied rewrites 74.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -1400000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;t\_1 \leq -1400000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -2e+72)
     (* t (/ y (- a z)))
     (if (<= t_1 -1400000000.0)
       (fma y (- 1.0 (/ t z)) x)
       (if (<= t_1 2e-22) t_2 (if (<= t_1 10000000.0) (+ 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 <= -2e+72) {
		tmp = t * (y / (a - z));
	} else if (t_1 <= -1400000000.0) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else if (t_1 <= 2e-22) {
		tmp = t_2;
	} else if (t_1 <= 10000000.0) {
		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 <= -2e+72)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	elseif (t_1 <= -1400000000.0)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	elseif (t_1 <= 2e-22)
		tmp = t_2;
	elseif (t_1 <= 10000000.0)
		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, -2e+72], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1400000000.0], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-22], t$95$2, If[LessEqual[t$95$1, 10000000.0], N[(x + y), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.99999999999999989e72

   1. Initial program 96.3%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 96.3

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

   4. Applied rewrites 96.3%

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

   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. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 z a)) < -1.4e9

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. 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 85.3

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

   5. Applied rewrites 85.3%

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

   if -1.4e9 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-22 or 1e7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.5%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

   if 2.0000000000000001e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e7

   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -1400000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;t\_1 \leq -1400000000:\\ \;\;\;\;\mathsf{fma}\left(y, -\frac{t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -2e+72)
     (* t (/ y (- a z)))
     (if (<= t_1 -1400000000.0)
       (fma y (- (/ t z)) x)
       (if (<= t_1 2e-22) t_2 (if (<= t_1 10000000.0) (+ 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 <= -2e+72) {
		tmp = t * (y / (a - z));
	} else if (t_1 <= -1400000000.0) {
		tmp = fma(y, -(t / z), x);
	} else if (t_1 <= 2e-22) {
		tmp = t_2;
	} else if (t_1 <= 10000000.0) {
		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 <= -2e+72)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	elseif (t_1 <= -1400000000.0)
		tmp = fma(y, Float64(-Float64(t / z)), x);
	elseif (t_1 <= 2e-22)
		tmp = t_2;
	elseif (t_1 <= 10000000.0)
		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, -2e+72], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1400000000.0], N[(y * (-N[(t / z), $MachinePrecision]) + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-22], t$95$2, If[LessEqual[t$95$1, 10000000.0], N[(x + y), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.99999999999999989e72

   1. Initial program 96.3%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 96.3

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

   4. Applied rewrites 96.3%

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

   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. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 z a)) < -1.4e9

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. 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 85.3

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

   5. Applied rewrites 85.3%

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

   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 83.7%

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

   if -1.4e9 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-22 or 1e7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.5%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

   if 2.0000000000000001e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e7

   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -1400000000:\\ \;\;\;\;\mathsf{fma}\left(y, -\frac{t}{z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;t\_1 \leq -1400000000:\\ \;\;\;\;\mathsf{fma}\left(y, -\frac{t}{z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -2e+72)
     (* y (/ t (- a z)))
     (if (<= t_1 -1400000000.0)
       (fma y (- (/ t z)) x)
       (if (<= t_1 2e-22) t_2 (if (<= t_1 10000000.0) (+ 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 <= -2e+72) {
		tmp = y * (t / (a - z));
	} else if (t_1 <= -1400000000.0) {
		tmp = fma(y, -(t / z), x);
	} else if (t_1 <= 2e-22) {
		tmp = t_2;
	} else if (t_1 <= 10000000.0) {
		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 <= -2e+72)
		tmp = Float64(y * Float64(t / Float64(a - z)));
	elseif (t_1 <= -1400000000.0)
		tmp = fma(y, Float64(-Float64(t / z)), x);
	elseif (t_1 <= 2e-22)
		tmp = t_2;
	elseif (t_1 <= 10000000.0)
		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, -2e+72], N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1400000000.0], N[(y * (-N[(t / z), $MachinePrecision]) + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-22], t$95$2, If[LessEqual[t$95$1, 10000000.0], N[(x + y), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.99999999999999989e72

   1. Initial program 96.3%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 84.2

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 84.5%

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

   if -1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 z a)) < -1.4e9

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. 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 85.3

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

   5. Applied rewrites 85.3%

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

   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 83.7%

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

   if -1.4e9 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-22 or 1e7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.5%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

   if 2.0000000000000001e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e7

   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -1400000000:\\ \;\;\;\;\mathsf{fma}\left(y, -\frac{t}{z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -1400000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 1.0000000001:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ t (- a z)) y x)))
   (if (<= t_1 -1400000000.0)
     t_2
     (if (<= t_1 2e-22)
       (fma y (/ (- t z) a) x)
       (if (<= t_1 1.0000000001) (fma (/ z (- z a)) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((t / (a - z)), y, x);
	double tmp;
	if (t_1 <= -1400000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-22) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 1.0000000001) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(t / Float64(a - z)), y, x)
	tmp = 0.0
	if (t_1 <= -1400000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e-22)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 1.0000000001)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1400000000.0], t$95$2, If[LessEqual[t$95$1, 2e-22], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1.0000000001], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.4e9 or 1.0000000001 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.9%

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

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if -1.4e9 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 2.0000000000000001e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000001

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 7: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -1400000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 1.0000000001:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ t (- a z)) y x)))
   (if (<= t_1 -1400000000.0)
     t_2
     (if (<= t_1 2e-22)
       (fma y (/ (- t z) a) x)
       (if (<= t_1 1.0000000001) (+ 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((t / (a - z)), y, x);
	double tmp;
	if (t_1 <= -1400000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-22) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 1.0000000001) {
		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(Float64(t / Float64(a - z)), y, x)
	tmp = 0.0
	if (t_1 <= -1400000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e-22)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 1.0000000001)
		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[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1400000000.0], t$95$2, If[LessEqual[t$95$1, 2e-22], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1.0000000001], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.4e9 or 1.0000000001 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.9%

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

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if -1.4e9 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 2.0000000000000001e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1400000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 1.0000000001:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 2e-22) t_2 (if (<= t_1 10000000.0) (+ 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 <= 2e-22) {
		tmp = t_2;
	} else if (t_1 <= 10000000.0) {
		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 <= 2e-22)
		tmp = t_2;
	elseif (t_1 <= 10000000.0)
		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, 2e-22], t$95$2, If[LessEqual[t$95$1, 10000000.0], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-22 or 1e7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.3%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   if 2.0000000000000001e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e7

   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+159}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -2e+72)
     (* y (/ t a))
     (if (<= t_1 5e+159) (+ x y) (* 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+72) {
		tmp = y * (t / a);
	} else if (t_1 <= 5e+159) {
		tmp = x + y;
	} else {
		tmp = t * (y / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    if (t_1 <= (-2d+72)) then
        tmp = y * (t / a)
    else if (t_1 <= 5d+159) then
        tmp = x + y
    else
        tmp = t * (y / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -2e+72) {
		tmp = y * (t / a);
	} else if (t_1 <= 5e+159) {
		tmp = x + y;
	} else {
		tmp = t * (y / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if t_1 <= -2e+72:
		tmp = y * (t / a)
	elif t_1 <= 5e+159:
		tmp = x + y
	else:
		tmp = 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+72)
		tmp = Float64(y * Float64(t / a));
	elseif (t_1 <= 5e+159)
		tmp = Float64(x + y);
	else
		tmp = Float64(t * Float64(y / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	tmp = 0.0;
	if (t_1 <= -2e+72)
		tmp = y * (t / a);
	elseif (t_1 <= 5e+159)
		tmp = x + y;
	else
		tmp = t * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+72], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+159], N[(x + y), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.99999999999999989e72

   1. Initial program 96.3%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 57.8%

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

   10. Applied rewrites 61.4%

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

   if -1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000003e159

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.8

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

   5. Applied rewrites 70.8%

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

   if 5.00000000000000003e159 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.5%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. 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 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 52.4%

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

   10. Applied rewrites 56.7%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+159}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := t \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+159}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* t (/ y a))))
   (if (<= t_1 -2e+72) t_2 (if (<= t_1 5e+159) (+ 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 = t * (y / a);
	double tmp;
	if (t_1 <= -2e+72) {
		tmp = t_2;
	} else if (t_1 <= 5e+159) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = t * (y / a)
    if (t_1 <= (-2d+72)) then
        tmp = t_2
    else if (t_1 <= 5d+159) then
        tmp = x + y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = t * (y / a);
	double tmp;
	if (t_1 <= -2e+72) {
		tmp = t_2;
	} else if (t_1 <= 5e+159) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = t * (y / a)
	tmp = 0
	if t_1 <= -2e+72:
		tmp = t_2
	elif t_1 <= 5e+159:
		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 = Float64(t * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -2e+72)
		tmp = t_2;
	elseif (t_1 <= 5e+159)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = t * (y / a);
	tmp = 0.0;
	if (t_1 <= -2e+72)
		tmp = t_2;
	elseif (t_1 <= 5e+159)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+72], t$95$2, If[LessEqual[t$95$1, 5e+159], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1.99999999999999989e72 or 5.00000000000000003e159 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.1%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 55.3%

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

   10. Applied rewrites 59.2%

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

   if -1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000003e159

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.8

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

   5. Applied rewrites 70.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+159}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 12: 98.1% 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 98.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 98.8

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

4. Applied rewrites 98.8%

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

Alternative 13: 95.9% 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 98.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. 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 95.8

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

4. Applied rewrites 95.8%

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

Alternative 14: 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 98.8%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 61.0

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

5. Applied rewrites 61.0%

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

6. Final simplification 61.0%

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

Developer Target 1: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024233 
(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)))))