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

Percentage Accurate: 98.0% → 98.0%

Time: 10.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 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 98.0%

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

Alternative 2: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := t \cdot \frac{y}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{-z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* t (/ y (- a z)))))
   (if (<= t_1 -2e+161)
     t_2
     (if (<= t_1 -5e+22)
       (fma y (/ t (- z)) x)
       (if (<= t_1 2e-95)
         (fma y (/ t a) x)
         (if (<= t_1 2e+84) (fma y (/ z (- z a)) 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 = t * (y / (a - z));
	double tmp;
	if (t_1 <= -2e+161) {
		tmp = t_2;
	} else if (t_1 <= -5e+22) {
		tmp = fma(y, (t / -z), x);
	} else if (t_1 <= 2e-95) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 2e+84) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(t * Float64(y / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -2e+161)
		tmp = t_2;
	elseif (t_1 <= -5e+22)
		tmp = fma(y, Float64(t / Float64(-z)), x);
	elseif (t_1 <= 2e-95)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 2e+84)
		tmp = fma(y, Float64(z / Float64(z - a)), 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[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+161], t$95$2, If[LessEqual[t$95$1, -5e+22], N[(y * N[(t / (-z)), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-95], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+84], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.0000000000000001e161 or 2.00000000000000012e84 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 88.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. neg-lowering-neg.f64 89.8

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

   5. Simplified 89.8%

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

      1. frac-2neg N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. unsub-neg N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-neg-out N/A

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

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

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

      10. frac-2neg N/A

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

      11. distribute-frac-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. remove-double-neg N/A

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 91.8

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

   7. Applied egg-rr 91.8%

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

   if -2.0000000000000001e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4.9999999999999996e22

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

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

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

      11. /-lowering-/.f64 84.9

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

   5. Simplified 84.9%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 84.9

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

   8. Simplified 84.9%

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

   if -4.9999999999999996e22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999998e-95

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 84.7

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

   5. Simplified 84.7%

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

   if 1.99999999999999998e-95 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000012e84

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 92.7

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

   5. Simplified 92.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+161}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{-z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := t \cdot \frac{y}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{-z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* t (/ y (- a z)))))
   (if (<= t_1 -2e+161)
     t_2
     (if (<= t_1 -5e+22)
       (fma y (/ t (- z)) x)
       (if (<= t_1 2e-10)
         (fma y (/ (- t z) a) x)
         (if (<= t_1 2e+84) (+ 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 - z));
	double tmp;
	if (t_1 <= -2e+161) {
		tmp = t_2;
	} else if (t_1 <= -5e+22) {
		tmp = fma(y, (t / -z), x);
	} else if (t_1 <= 2e-10) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 2e+84) {
		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 / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -2e+161)
		tmp = t_2;
	elseif (t_1 <= -5e+22)
		tmp = fma(y, Float64(t / Float64(-z)), x);
	elseif (t_1 <= 2e-10)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 2e+84)
		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[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+161], t$95$2, If[LessEqual[t$95$1, -5e+22], N[(y * N[(t / (-z)), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-10], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+84], N[(x + y), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.0000000000000001e161 or 2.00000000000000012e84 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 88.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. neg-lowering-neg.f64 89.8

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

   5. Simplified 89.8%

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

      1. frac-2neg N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. unsub-neg N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-neg-out N/A

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

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

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

      10. frac-2neg N/A

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

      11. distribute-frac-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. remove-double-neg N/A

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 91.8

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

   7. Applied egg-rr 91.8%

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

   if -2.0000000000000001e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4.9999999999999996e22

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

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

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

      11. /-lowering-/.f64 84.9

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

   5. Simplified 84.9%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 84.9

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

   8. Simplified 84.9%

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

   if -4.9999999999999996e22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000007e-10

   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. accelerator-lowering-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. /-lowering-/.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. --lowering--.f64 96.2

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

   5. Simplified 96.2%

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

   if 2.00000000000000007e-10 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000012e84

   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. +-lowering-+.f64 92.9

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

   5. Simplified 92.9%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+161}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{-z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := t \cdot \frac{y}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{-z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* t (/ y (- a z)))))
   (if (<= t_1 -2e+161)
     t_2
     (if (<= t_1 -5e+22)
       (fma y (/ t (- z)) x)
       (if (<= t_1 1e-28)
         (fma y (/ t a) x)
         (if (<= t_1 2e+84) (+ 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 - z));
	double tmp;
	if (t_1 <= -2e+161) {
		tmp = t_2;
	} else if (t_1 <= -5e+22) {
		tmp = fma(y, (t / -z), x);
	} else if (t_1 <= 1e-28) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 2e+84) {
		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 / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -2e+161)
		tmp = t_2;
	elseif (t_1 <= -5e+22)
		tmp = fma(y, Float64(t / Float64(-z)), x);
	elseif (t_1 <= 1e-28)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 2e+84)
		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[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+161], t$95$2, If[LessEqual[t$95$1, -5e+22], N[(y * N[(t / (-z)), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-28], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+84], N[(x + y), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.0000000000000001e161 or 2.00000000000000012e84 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 88.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. neg-lowering-neg.f64 89.8

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

   5. Simplified 89.8%

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

      1. frac-2neg N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. unsub-neg N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-neg-out N/A

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

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

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

      10. frac-2neg N/A

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

      11. distribute-frac-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. remove-double-neg N/A

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 91.8

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

   7. Applied egg-rr 91.8%

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

   if -2.0000000000000001e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4.9999999999999996e22

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

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

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

      11. /-lowering-/.f64 84.9

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

   5. Simplified 84.9%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 84.9

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

   8. Simplified 84.9%

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

   if -4.9999999999999996e22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.99999999999999971e-29

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 82.0

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

   5. Simplified 82.0%

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

   if 9.99999999999999971e-29 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000012e84

   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. +-lowering-+.f64 91.6

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

   5. Simplified 91.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+161}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{-z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ y (- z a)) (- t) x)))
   (if (<= t_1 -2.0)
     t_2
     (if (<= t_1 2e-10)
       (fma y (/ (- t z) a) x)
       (if (<= t_1 2.0) (+ x (* y (- 1.0 (/ t z)))) 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 / (z - a)), -t, x);
	double tmp;
	if (t_1 <= -2.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-10) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 2.0) {
		tmp = x + (y * (1.0 - (t / z)));
	} 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(y / Float64(z - a)), Float64(-t), x)
	tmp = 0.0
	if (t_1 <= -2.0)
		tmp = t_2;
	elseif (t_1 <= 2e-10)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x + Float64(y * Float64(1.0 - Float64(t / z))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * (-t) + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2.0], t$95$2, If[LessEqual[t$95$1, 2e-10], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x + N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. --lowering--.f64 94.5

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

   4. Applied egg-rr 94.5%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 94.5

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

   7. Simplified 94.5%

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

   if -2 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000007e-10

   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. accelerator-lowering-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. /-lowering-/.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. --lowering--.f64 98.4

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

   5. Simplified 98.4%

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

   if 2.00000000000000007e-10 < (/.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 a around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

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

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

      4. /-lowering-/.f64 99.9

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

   5. Simplified 99.9%

      \[\leadsto x + y \cdot \color{blue}{\left(1 - \frac{t}{z}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := t \cdot \frac{y}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* t (/ y (- a z)))))
   (if (<= t_1 -2e+138)
     t_2
     (if (<= t_1 1e-28) (fma y (/ t a) x) (if (<= t_1 2e+84) (+ 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 - z));
	double tmp;
	if (t_1 <= -2e+138) {
		tmp = t_2;
	} else if (t_1 <= 1e-28) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 2e+84) {
		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 / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -2e+138)
		tmp = t_2;
	elseif (t_1 <= 1e-28)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 2e+84)
		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[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+138], t$95$2, If[LessEqual[t$95$1, 1e-28], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+84], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.0000000000000001e138 or 2.00000000000000012e84 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 90.1%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. neg-lowering-neg.f64 85.2

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

   5. Simplified 85.2%

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

      1. frac-2neg N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. unsub-neg N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-neg-out N/A

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

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

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

      10. frac-2neg N/A

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

      11. distribute-frac-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. remove-double-neg N/A

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 87.0

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

   7. Applied egg-rr 87.0%

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

   if -2.0000000000000001e138 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.99999999999999971e-29

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 78.3

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

   5. Simplified 78.3%

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

   if 9.99999999999999971e-29 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000012e84

   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. +-lowering-+.f64 91.6

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

   5. Simplified 91.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+138}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := t \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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+161)
     t_2
     (if (<= t_1 5e-31) x (if (<= t_1 4e+84) (+ 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+161) {
		tmp = t_2;
	} else if (t_1 <= 5e-31) {
		tmp = x;
	} else if (t_1 <= 4e+84) {
		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+161)) then
        tmp = t_2
    else if (t_1 <= 5d-31) then
        tmp = x
    else if (t_1 <= 4d+84) 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+161) {
		tmp = t_2;
	} else if (t_1 <= 5e-31) {
		tmp = x;
	} else if (t_1 <= 4e+84) {
		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+161:
		tmp = t_2
	elif t_1 <= 5e-31:
		tmp = x
	elif t_1 <= 4e+84:
		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+161)
		tmp = t_2;
	elseif (t_1 <= 5e-31)
		tmp = x;
	elseif (t_1 <= 4e+84)
		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+161)
		tmp = t_2;
	elseif (t_1 <= 5e-31)
		tmp = x;
	elseif (t_1 <= 4e+84)
		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+161], t$95$2, If[LessEqual[t$95$1, 5e-31], x, If[LessEqual[t$95$1, 4e+84], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.0000000000000001e161 or 4.00000000000000023e84 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 88.7%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. neg-lowering-neg.f64 89.6

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

   5. Simplified 89.6%

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

      1. frac-2neg N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. unsub-neg N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. distribute-rgt-neg-out N/A

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

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

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

      10. frac-2neg N/A

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

      11. distribute-frac-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. remove-double-neg N/A

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 91.6

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

   7. Applied egg-rr 91.6%

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

   8. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 61.1

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

   10. Simplified 61.1%

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

   if -2.0000000000000001e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e-31

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.8%

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

   if 5e-31 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000023e84

   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. +-lowering-+.f64 90.2

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

   5. Simplified 90.2%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+161}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 4 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot \frac{t}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y (/ t a))))
   (if (<= t_1 -2e+161)
     t_2
     (if (<= t_1 5e-31) x (if (<= t_1 4e+84) (+ 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 = y * (t / a);
	double tmp;
	if (t_1 <= -2e+161) {
		tmp = t_2;
	} else if (t_1 <= 5e-31) {
		tmp = x;
	} else if (t_1 <= 4e+84) {
		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 = y * (t / a)
    if (t_1 <= (-2d+161)) then
        tmp = t_2
    else if (t_1 <= 5d-31) then
        tmp = x
    else if (t_1 <= 4d+84) 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 = y * (t / a);
	double tmp;
	if (t_1 <= -2e+161) {
		tmp = t_2;
	} else if (t_1 <= 5e-31) {
		tmp = x;
	} else if (t_1 <= 4e+84) {
		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 = y * (t / a)
	tmp = 0
	if t_1 <= -2e+161:
		tmp = t_2
	elif t_1 <= 5e-31:
		tmp = x
	elif t_1 <= 4e+84:
		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(y * Float64(t / a))
	tmp = 0.0
	if (t_1 <= -2e+161)
		tmp = t_2;
	elseif (t_1 <= 5e-31)
		tmp = x;
	elseif (t_1 <= 4e+84)
		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 = y * (t / a);
	tmp = 0.0;
	if (t_1 <= -2e+161)
		tmp = t_2;
	elseif (t_1 <= 5e-31)
		tmp = x;
	elseif (t_1 <= 4e+84)
		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[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+161], t$95$2, If[LessEqual[t$95$1, 5e-31], x, If[LessEqual[t$95$1, 4e+84], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2.0000000000000001e161 or 4.00000000000000023e84 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 88.7%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. accelerator-lowering-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. /-lowering-/.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. --lowering--.f64 58.8

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

   5. Simplified 58.8%

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

   6. Taylor expanded in y around inf

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

      1. div-sub N/A

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

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

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

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

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

      4. --lowering--.f64 56.6

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

   8. Simplified 56.6%

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

   9. Taylor expanded in t around inf

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

       1. /-lowering-/.f64 56.6

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

   11. Simplified 56.6%

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

   if -2.0000000000000001e161 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e-31

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.8%

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

   if 5e-31 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000023e84

   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. +-lowering-+.f64 90.2

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

   5. Simplified 90.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+161}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 4 \cdot 10^{+84}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ y (- z a)) (- z t) x)))
   (if (<= t_1 2e-95) t_2 (if (<= t_1 1.0) (fma y (/ z (- z a)) 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((y / (z - a)), (z - t), x);
	double tmp;
	if (t_1 <= 2e-95) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = fma(y, (z / (z - a)), 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(y / Float64(z - a)), Float64(z - t), x)
	tmp = 0.0
	if (t_1 <= 2e-95)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-95], t$95$2, If[LessEqual[t$95$1, 1.0], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999998e-95 or 1 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.7%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. --lowering--.f64 96.7

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

   4. Applied egg-rr 96.7%

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

   if 1.99999999999999998e-95 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      5. --lowering--.f64 99.0

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

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 10^{-28}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 1e-28) t_2 (if (<= t_1 10.0) (+ 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(t, (y / a), x);
	double tmp;
	if (t_1 <= 1e-28) {
		tmp = t_2;
	} else if (t_1 <= 10.0) {
		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(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 1e-28)
		tmp = t_2;
	elseif (t_1 <= 10.0)
		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[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-28], t$95$2, If[LessEqual[t$95$1, 10.0], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.99999999999999971e-29 or 10 < (/.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. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. --lowering--.f64 96.3

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

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 89.5

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

   7. Simplified 89.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   9. 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. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 70.7

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

   10. Simplified 70.7%

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

   if 9.99999999999999971e-29 < (/.f64 (-.f64 z t) (-.f64 z a)) < 10

   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. +-lowering-+.f64 96.7

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

   5. Simplified 96.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+172}:\\ \;\;\;\;y\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (<= t_1 -2e+172) y (if (<= t_1 1e-5) x y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -2e+172) {
		tmp = y;
	} else if (t_1 <= 1e-5) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * ((z - t) / (z - a))
    if (t_1 <= (-2d+172)) then
        tmp = y
    else if (t_1 <= 1d-5) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -2e+172) {
		tmp = y;
	} else if (t_1 <= 1e-5) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (z - a))
	tmp = 0
	if t_1 <= -2e+172:
		tmp = y
	elif t_1 <= 1e-5:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if (t_1 <= -2e+172)
		tmp = y;
	elseif (t_1 <= 1e-5)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (z - a));
	tmp = 0.0;
	if (t_1 <= -2e+172)
		tmp = y;
	elseif (t_1 <= 1e-5)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+172], y, If[LessEqual[t$95$1, 1e-5], x, y]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -2.0000000000000002e172 or 1.00000000000000008e-5 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.0

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

   5. Simplified 42.0%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 35.2%

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

   if -2.0000000000000002e172 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 1.00000000000000008e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 70.0%

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

Alternative 12: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 6.4 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 6.4e-31) x (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 6.4e-31) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((z - t) / (z - a)) <= 6.4d-31) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 6.4e-31) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 6.4e-31:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 6.4e-31)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (z - a)) <= 6.4e-31)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 6.4e-31], x, N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 6.40000000000000036e-31

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 54.6%

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

   if 6.40000000000000036e-31 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.3%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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. +-lowering-+.f64 74.7

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

   5. Simplified 74.7%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 6.4 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.8% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around inf

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

5. Simplified 45.8%

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

Developer Target 1: 98.2% 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 2024205 
(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)))))