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

Percentage Accurate: 98.1% → 98.1%

Time: 10.4s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. --lowering--.f64 N/A

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

   6. --lowering--.f64 99.2

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

4. Applied egg-rr 99.2%

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

Alternative 2: 83.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ t_2 := \frac{z - t}{z - a}\\ t_3 := y \cdot \frac{t}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+61}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{elif}\;t\_2 \leq 2.5 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ t z)) x))
        (t_2 (/ (- z t) (- z a)))
        (t_3 (* y (/ t (- a z)))))
   (if (<= t_2 -1e+61)
     t_3
     (if (<= t_2 -5000.0)
       t_1
       (if (<= t_2 -4e-183)
         (fma y (/ t a) x)
         (if (<= t_2 0.2)
           (- x (/ (* z y) a))
           (if (<= t_2 2.5e+47) t_1 t_3)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (t / z)), x);
	double t_2 = (z - t) / (z - a);
	double t_3 = y * (t / (a - z));
	double tmp;
	if (t_2 <= -1e+61) {
		tmp = t_3;
	} else if (t_2 <= -5000.0) {
		tmp = t_1;
	} else if (t_2 <= -4e-183) {
		tmp = fma(y, (t / a), x);
	} else if (t_2 <= 0.2) {
		tmp = x - ((z * y) / a);
	} else if (t_2 <= 2.5e+47) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(t / z)), x)
	t_2 = Float64(Float64(z - t) / Float64(z - a))
	t_3 = Float64(y * Float64(t / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= -1e+61)
		tmp = t_3;
	elseif (t_2 <= -5000.0)
		tmp = t_1;
	elseif (t_2 <= -4e-183)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_2 <= 0.2)
		tmp = Float64(x - Float64(Float64(z * y) / a));
	elseif (t_2 <= 2.5e+47)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+61], t$95$3, If[LessEqual[t$95$2, -5000.0], t$95$1, If[LessEqual[t$95$2, -4e-183], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(x - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.5e+47], t$95$1, t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -9.99999999999999949e60 or 2.50000000000000011e47 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.7%

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

   3. Taylor expanded in 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 67.6

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

   5. Simplified 67.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 75.3

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

   7. Applied egg-rr 75.3%

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

   if -9.99999999999999949e60 < (/.f64 (-.f64 z t) (-.f64 z a)) < -5e3 or 0.20000000000000001 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.50000000000000011e47

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

      7. 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 98.0

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

   5. Simplified 98.0%

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

   if -5e3 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4.00000000000000002e-183

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-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 90.0

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

   5. Simplified 90.0%

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

   if -4.00000000000000002e-183 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.20000000000000001

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 98.2

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 93.6%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. *-lowering-*.f64 90.5

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

   10. Simplified 90.5%

       \[\leadsto \color{blue}{x - \frac{y \cdot z}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0.2:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2.5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot \frac{t}{a - z}\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{elif}\;t\_1 \leq 4000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y (/ t (- a z)))))
   (if (<= t_1 -5000.0)
     t_2
     (if (<= t_1 -4e-183)
       (fma y (/ t a) x)
       (if (<= t_1 0.2)
         (- x (/ (* z y) a))
         (if (<= t_1 4000000000000.0) (+ y x) t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * (t / (a - z));
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= -4e-183) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 0.2) {
		tmp = x - ((z * y) / a);
	} else if (t_1 <= 4000000000000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(y * Float64(t / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= -4e-183)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 0.2)
		tmp = Float64(x - Float64(Float64(z * y) / a));
	elseif (t_1 <= 4000000000000.0)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$2, If[LessEqual[t$95$1, -4e-183], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.2], N[(x - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4000000000000.0], N[(y + x), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5e3 or 4e12 < (/.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 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 61.2

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

   5. Simplified 61.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 69.9

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

   7. Applied egg-rr 69.9%

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

   if -5e3 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4.00000000000000002e-183

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-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 90.0

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

   5. Simplified 90.0%

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

   if -4.00000000000000002e-183 < (/.f64 (-.f64 z t) (-.f64 z a)) < 0.20000000000000001

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 98.2

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 93.6%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. *-lowering-*.f64 90.5

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

   10. Simplified 90.5%

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

   if 0.20000000000000001 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4e12

   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 98.7

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

   5. Simplified 98.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5000:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0.2:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 4000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot \frac{t}{a - z}\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 4000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y (/ t (- a z)))))
   (if (<= t_1 -5000.0)
     t_2
     (if (<= t_1 2e-26)
       (fma y (/ t a) x)
       (if (<= t_1 4000000000000.0) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * (t / (a - z));
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-26) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 4000000000000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(y * Float64(t / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 2e-26)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 4000000000000.0)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$2, If[LessEqual[t$95$1, 2e-26], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 4000000000000.0], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5e3 or 4e12 < (/.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 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 61.2

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

   5. Simplified 61.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 69.9

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

   7. Applied egg-rr 69.9%

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

   if -5e3 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-26

   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 85.1

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

   5. Simplified 85.1%

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

   if 2.0000000000000001e-26 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4e12

   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 94.3

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

   5. Simplified 94.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5000:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 4000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot \frac{t}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+216}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y (/ t a))))
   (if (<= t_1 -2e+216)
     t_2
     (if (<= t_1 1e-40) x (if (<= t_1 1e+51) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = y * (t / a);
	double tmp;
	if (t_1 <= -2e+216) {
		tmp = t_2;
	} else if (t_1 <= 1e-40) {
		tmp = x;
	} else if (t_1 <= 1e+51) {
		tmp = y + x;
	} 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+216)) then
        tmp = t_2
    else if (t_1 <= 1d-40) then
        tmp = x
    else if (t_1 <= 1d+51) then
        tmp = y + x
    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+216) {
		tmp = t_2;
	} else if (t_1 <= 1e-40) {
		tmp = x;
	} else if (t_1 <= 1e+51) {
		tmp = y + x;
	} 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+216:
		tmp = t_2
	elif t_1 <= 1e-40:
		tmp = x
	elif t_1 <= 1e+51:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(y * Float64(t / a))
	tmp = 0.0
	if (t_1 <= -2e+216)
		tmp = t_2;
	elseif (t_1 <= 1e-40)
		tmp = x;
	elseif (t_1 <= 1e+51)
		tmp = Float64(y + x);
	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+216)
		tmp = t_2;
	elseif (t_1 <= 1e-40)
		tmp = x;
	elseif (t_1 <= 1e+51)
		tmp = y + x;
	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+216], t$95$2, If[LessEqual[t$95$1, 1e-40], x, If[LessEqual[t$95$1, 1e+51], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e216 or 1e51 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 80.6

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

   5. Simplified 80.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 83.2

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

   7. Applied egg-rr 83.2%

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

   8. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 60.9

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

   10. Simplified 60.9%

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

   if -2e216 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999993e-41

   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 64.0%

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

   if 9.9999999999999993e-41 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e51

   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 91.3

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

   5. Simplified 91.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+216}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := y \cdot \frac{t}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (* y (/ t (- a z)))))
   (if (<= t_1 -2e+23)
     t_2
     (if (<= t_1 4000000000000.0) (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 = y * (t / (a - z));
	double tmp;
	if (t_1 <= -2e+23) {
		tmp = t_2;
	} else if (t_1 <= 4000000000000.0) {
		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 = Float64(y * Float64(t / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -2e+23)
		tmp = t_2;
	elseif (t_1 <= 4000000000000.0)
		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[(y * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+23], t$95$2, If[LessEqual[t$95$1, 4000000000000.0], 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)) < -1.9999999999999998e23 or 4e12 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.1%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 64.8

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

   5. Simplified 64.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 74.2

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

   7. Applied egg-rr 74.2%

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

   if -1.9999999999999998e23 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4e12

   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 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -2 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 4000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2.5 \cdot 10^{+47}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma y (/ t a) x)))
   (if (<= t_1 2e-26) t_2 (if (<= t_1 2.5e+47) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(y, (t / a), x);
	double tmp;
	if (t_1 <= 2e-26) {
		tmp = t_2;
	} else if (t_1 <= 2.5e+47) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(y, Float64(t / a), x)
	tmp = 0.0
	if (t_1 <= 2e-26)
		tmp = t_2;
	elseif (t_1 <= 2.5e+47)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-26], t$95$2, If[LessEqual[t$95$1, 2.5e+47], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-26 or 2.50000000000000011e47 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-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 71.6

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

   5. Simplified 71.6%

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

   if 2.0000000000000001e-26 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.50000000000000011e47

   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 92.7

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

   5. Simplified 92.7%

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

Alternative 8: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+25}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 2e-26) t_2 (if (<= t_1 2e+25) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(t, (y / a), x);
	double tmp;
	if (t_1 <= 2e-26) {
		tmp = t_2;
	} else if (t_1 <= 2e+25) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 2e-26)
		tmp = t_2;
	elseif (t_1 <= 2e+25)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-26], t$95$2, If[LessEqual[t$95$1, 2e+25], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.0000000000000001e-26 or 2.00000000000000018e25 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 67.8

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

   5. Simplified 67.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 70.1

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

   7. Applied egg-rr 70.1%

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

   if 2.0000000000000001e-26 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000018e25

   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 93.5

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

   5. Simplified 93.5%

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

Alternative 9: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 1.8 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 1.8e-40) x (+ y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 1.8e-40) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((z - t) / (z - a)) <= 1.8d-40) then
        tmp = x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 1.8e-40) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 1.8e-40:
		tmp = x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 1.8e-40)
		tmp = x;
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (z - a)) <= 1.8e-40)
		tmp = x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 1.8e-40], x, N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.8e-40

   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 60.8%

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

   if 1.8e-40 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 74.4

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

   5. Simplified 74.4%

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

Alternative 10: 95.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[x + y \cdot \frac{z - t}{z - a} \]
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 94.6

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

4. Applied egg-rr 94.6%

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

Alternative 11: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-146}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-100}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.9e-146) x (if (<= x 6.5e-100) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.9e-146) {
		tmp = x;
	} else if (x <= 6.5e-100) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-1.9d-146)) then
        tmp = x
    else if (x <= 6.5d-100) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.9e-146) {
		tmp = x;
	} else if (x <= 6.5e-100) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.9e-146:
		tmp = x
	elif x <= 6.5e-100:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.9e-146)
		tmp = x;
	elseif (x <= 6.5e-100)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.9e-146)
		tmp = x;
	elseif (x <= 6.5e-100)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.9e-146], x, If[LessEqual[x, 6.5e-100], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.89999999999999997e-146 or 6.50000000000000013e-100 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 65.9%

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

   if -1.89999999999999997e-146 < x < 6.50000000000000013e-100

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 49.6

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

   5. Simplified 49.6%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 43.7%

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

Alternative 12: 50.1% 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 99.2%

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

3. Taylor expanded in x around inf

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

5. Simplified 49.1%

   \[\leadsto \color{blue}{x} \]
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 2024199 
(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)))))