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

Percentage Accurate: 98.3% → 96.1%

Time: 11.1s

Alternatives: 17

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: 98.3% 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: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ t_2 := \mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))) (t_2 (fma (/ y (- z a)) (- z t) x)))
   (if (<= t_1 4e-17) t_2 (if (<= t_1 5e+39) (fma y (- 1.0 (/ t z)) x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double t_2 = fma((y / (z - a)), (z - t), x);
	double tmp;
	if (t_1 <= 4e-17) {
		tmp = t_2;
	} else if (t_1 <= 5e+39) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	t_2 = fma(Float64(y / Float64(z - a)), Float64(z - t), x)
	tmp = 0.0
	if (t_1 <= 4e-17)
		tmp = t_2;
	elseif (t_1 <= 5e+39)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(a - z), $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, 4e-17], t$95$2, If[LessEqual[t$95$1, 5e+39], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000029e-17 or 5.00000000000000015e39 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.1%

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

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

   if 4.00000000000000029e-17 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000015e39

   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 99.6

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-a}, z, x\right)\\ \mathbf{elif}\;t\_1 \leq 1.00000001:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))))
   (if (<= t_1 -2e+65)
     (fma (/ y z) (- t) x)
     (if (<= t_1 -4e-172)
       (fma y (/ t a) x)
       (if (<= t_1 5e-22)
         (fma (/ y (- a)) z x)
         (if (<= t_1 1.00000001) (+ y x) (- x (/ (* t y) z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double tmp;
	if (t_1 <= -2e+65) {
		tmp = fma((y / z), -t, x);
	} else if (t_1 <= -4e-172) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 5e-22) {
		tmp = fma((y / -a), z, x);
	} else if (t_1 <= 1.00000001) {
		tmp = y + x;
	} else {
		tmp = x - ((t * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -2e+65)
		tmp = fma(Float64(y / z), Float64(-t), x);
	elseif (t_1 <= -4e-172)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 5e-22)
		tmp = fma(Float64(y / Float64(-a)), z, x);
	elseif (t_1 <= 1.00000001)
		tmp = Float64(y + x);
	else
		tmp = Float64(x - Float64(Float64(t * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+65], N[(N[(y / z), $MachinePrecision] * (-t) + x), $MachinePrecision], If[LessEqual[t$95$1, -4e-172], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e-22], N[(N[(y / (-a)), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t$95$1, 1.00000001], N[(y + x), $MachinePrecision], N[(x - N[(N[(t * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e65

   1. Initial program 91.5%

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 72.8%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 72.8

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

   10. Simplified 72.8%

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

   if -2e65 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4.0000000000000002e-172

   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 87.3

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

   5. Simplified 87.3%

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

   if -4.0000000000000002e-172 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999954e-22

   1. Initial program 99.1%

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

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

   4. Applied egg-rr 100.0%

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

   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 98.3%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 98.3

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

   10. Simplified 98.3%

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

   if 4.99999999999999954e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000099999999

   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 99.0

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

   5. Simplified 99.0%

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

   if 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 91.5

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

   7. Simplified 91.5%

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

   8. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. *-lowering-*.f64 63.6

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

   10. Simplified 63.6%

       \[\leadsto \color{blue}{x - \frac{t \cdot y}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-a}, z, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 1.00000001:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{elif}\;t\_1 \leq 1.00000001:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))))
   (if (<= t_1 -2e+65)
     (fma (/ y z) (- t) x)
     (if (<= t_1 -4e-172)
       (fma y (/ t a) x)
       (if (<= t_1 5e-22)
         (- x (/ (* z y) a))
         (if (<= t_1 1.00000001) (+ y x) (- x (/ (* t y) z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double tmp;
	if (t_1 <= -2e+65) {
		tmp = fma((y / z), -t, x);
	} else if (t_1 <= -4e-172) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 5e-22) {
		tmp = x - ((z * y) / a);
	} else if (t_1 <= 1.00000001) {
		tmp = y + x;
	} else {
		tmp = x - ((t * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -2e+65)
		tmp = fma(Float64(y / z), Float64(-t), x);
	elseif (t_1 <= -4e-172)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 5e-22)
		tmp = Float64(x - Float64(Float64(z * y) / a));
	elseif (t_1 <= 1.00000001)
		tmp = Float64(y + x);
	else
		tmp = Float64(x - Float64(Float64(t * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+65], N[(N[(y / z), $MachinePrecision] * (-t) + x), $MachinePrecision], If[LessEqual[t$95$1, -4e-172], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e-22], N[(x - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.00000001], N[(y + x), $MachinePrecision], N[(x - N[(N[(t * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e65

   1. Initial program 91.5%

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 72.8%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 72.8

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

   10. Simplified 72.8%

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

   if -2e65 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4.0000000000000002e-172

   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 87.3

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

   5. Simplified 87.3%

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

   if -4.0000000000000002e-172 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999954e-22

   1. Initial program 99.1%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot z}{a}} \]
   7. 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 98.3

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

   8. Simplified 98.3%

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

   if 4.99999999999999954e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000099999999

   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 99.0

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

   5. Simplified 99.0%

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

   if 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 91.5

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

   7. Simplified 91.5%

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

   8. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. *-lowering-*.f64 63.6

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

   10. Simplified 63.6%

       \[\leadsto \color{blue}{x - \frac{t \cdot y}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 1.00000001:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ t_2 := x - \frac{t \cdot y}{z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+221}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{elif}\;t\_1 \leq 1.00000001:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))) (t_2 (- x (/ (* t y) z))))
   (if (<= t_1 -1e+221)
     t_2
     (if (<= t_1 -4e-172)
       (fma y (/ t a) x)
       (if (<= t_1 5e-22)
         (- x (/ (* z y) a))
         (if (<= t_1 1.00000001) (+ y x) t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double t_2 = x - ((t * y) / z);
	double tmp;
	if (t_1 <= -1e+221) {
		tmp = t_2;
	} else if (t_1 <= -4e-172) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 5e-22) {
		tmp = x - ((z * y) / a);
	} else if (t_1 <= 1.00000001) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	t_2 = Float64(x - Float64(Float64(t * y) / z))
	tmp = 0.0
	if (t_1 <= -1e+221)
		tmp = t_2;
	elseif (t_1 <= -4e-172)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 5e-22)
		tmp = Float64(x - Float64(Float64(z * y) / a));
	elseif (t_1 <= 1.00000001)
		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[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(t * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+221], t$95$2, If[LessEqual[t$95$1, -4e-172], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e-22], N[(x - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.00000001], N[(y + x), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e221 or 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 89.2%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 89.2

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 88.1

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

   7. Simplified 88.1%

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

   8. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. *-lowering-*.f64 68.8

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

   10. Simplified 68.8%

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

   if -1e221 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4.0000000000000002e-172

   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 77.6

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

   5. Simplified 77.6%

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

   if -4.0000000000000002e-172 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999954e-22

   1. Initial program 99.1%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot z}{a}} \]
   7. 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 98.3

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

   8. Simplified 98.3%

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

   if 4.99999999999999954e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000099999999

   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 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -1 \cdot 10^{+221}:\\ \;\;\;\;x - \frac{t \cdot y}{z}\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 1.00000001:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ t_2 := \mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))) (t_2 (fma (/ t (- a z)) y x)))
   (if (<= t_1 -2e+65)
     t_2
     (if (<= t_1 4e-17)
       (fma y (/ (- t z) a) x)
       (if (<= t_1 10000000000000.0) (fma y (- 1.0 (/ t z)) x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double t_2 = fma((t / (a - z)), y, x);
	double tmp;
	if (t_1 <= -2e+65) {
		tmp = t_2;
	} else if (t_1 <= 4e-17) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 10000000000000.0) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	t_2 = fma(Float64(t / Float64(a - z)), y, x)
	tmp = 0.0
	if (t_1 <= -2e+65)
		tmp = t_2;
	elseif (t_1 <= 4e-17)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 10000000000000.0)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+65], t$95$2, If[LessEqual[t$95$1, 4e-17], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 10000000000000.0], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e65 or 1e13 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 92.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 92.0

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 92.0

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

   7. Simplified 92.0%

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

   if -2e65 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000029e-17

   1. Initial program 99.4%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.0

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

   5. Simplified 99.0%

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

   if 4.00000000000000029e-17 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e13

   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 99.6

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 10000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))))
   (if (<= t_1 -2e+65)
     (fma (/ y z) (- t) x)
     (if (<= t_1 4e-17)
       (fma y (/ (- t z) a) x)
       (if (<= t_1 4e+55) (fma y (- 1.0 (/ t z)) x) (/ (* t y) (- a z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double tmp;
	if (t_1 <= -2e+65) {
		tmp = fma((y / z), -t, x);
	} else if (t_1 <= 4e-17) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 4e+55) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else {
		tmp = (t * y) / (a - z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -2e+65)
		tmp = fma(Float64(y / z), Float64(-t), x);
	elseif (t_1 <= 4e-17)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 4e+55)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	else
		tmp = Float64(Float64(t * y) / Float64(a - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+65], N[(N[(y / z), $MachinePrecision] * (-t) + x), $MachinePrecision], If[LessEqual[t$95$1, 4e-17], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+55], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e65

   1. Initial program 91.5%

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 72.8%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 72.8

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

   10. Simplified 72.8%

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

   if -2e65 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000029e-17

   1. Initial program 99.4%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.0

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

   5. Simplified 99.0%

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

   if 4.00000000000000029e-17 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000004e55

   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 97.6

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

   5. Simplified 97.6%

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

   if 4.00000000000000004e55 < (/.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 73.8

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

   5. Simplified 73.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))))
   (if (<= t_1 -2e+65)
     (fma (/ y z) (- t) x)
     (if (<= t_1 -4e-172)
       (fma y (/ t a) x)
       (if (<= t_1 4e-17) (fma (/ y (- a)) z x) (fma y (- 1.0 (/ t z)) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double tmp;
	if (t_1 <= -2e+65) {
		tmp = fma((y / z), -t, x);
	} else if (t_1 <= -4e-172) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 4e-17) {
		tmp = fma((y / -a), z, x);
	} else {
		tmp = fma(y, (1.0 - (t / z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -2e+65)
		tmp = fma(Float64(y / z), Float64(-t), x);
	elseif (t_1 <= -4e-172)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 4e-17)
		tmp = fma(Float64(y / Float64(-a)), z, x);
	else
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+65], N[(N[(y / z), $MachinePrecision] * (-t) + x), $MachinePrecision], If[LessEqual[t$95$1, -4e-172], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 4e-17], N[(N[(y / (-a)), $MachinePrecision] * z + x), $MachinePrecision], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e65

   1. Initial program 91.5%

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 72.8%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 72.8

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

   10. Simplified 72.8%

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

   if -2e65 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4.0000000000000002e-172

   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 87.3

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

   5. Simplified 87.3%

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

   if -4.0000000000000002e-172 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000029e-17

   1. Initial program 99.1%

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

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

   4. Applied egg-rr 100.0%

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

   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 98.4%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 98.4

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

   10. Simplified 98.4%

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

   if 4.00000000000000029e-17 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 86.9

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

   5. Simplified 86.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ t_2 := x - \frac{t \cdot y}{z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+221}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 1.00000001:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))) (t_2 (- x (/ (* t y) z))))
   (if (<= t_1 -1e+221)
     t_2
     (if (<= t_1 5e-22)
       (fma y (/ t a) x)
       (if (<= t_1 1.00000001) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double t_2 = x - ((t * y) / z);
	double tmp;
	if (t_1 <= -1e+221) {
		tmp = t_2;
	} else if (t_1 <= 5e-22) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 1.00000001) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	t_2 = Float64(x - Float64(Float64(t * y) / z))
	tmp = 0.0
	if (t_1 <= -1e+221)
		tmp = t_2;
	elseif (t_1 <= 5e-22)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 1.00000001)
		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[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(t * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+221], t$95$2, If[LessEqual[t$95$1, 5e-22], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1.00000001], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e221 or 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 89.2%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 89.2

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 88.1

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

   7. Simplified 88.1%

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

   8. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. *-lowering-*.f64 68.8

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

   10. Simplified 68.8%

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

   if -1e221 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999954e-22

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 84.0

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

   5. Simplified 84.0%

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

   if 4.99999999999999954e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000099999999

   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 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -1 \cdot 10^{+221}:\\ \;\;\;\;x - \frac{t \cdot y}{z}\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 1.00000001:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+221}:\\ \;\;\;\;-\frac{t \cdot y}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+39}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))))
   (if (<= t_1 -1e+221)
     (- (/ (* t y) z))
     (if (<= t_1 5e-22)
       (fma y (/ t a) x)
       (if (<= t_1 5e+39) (+ y x) (fma t (/ y a) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double tmp;
	if (t_1 <= -1e+221) {
		tmp = -((t * y) / z);
	} else if (t_1 <= 5e-22) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 5e+39) {
		tmp = y + x;
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -1e+221)
		tmp = Float64(-Float64(Float64(t * y) / z));
	elseif (t_1 <= 5e-22)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 5e+39)
		tmp = Float64(y + x);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+221], (-N[(N[(t * y), $MachinePrecision] / z), $MachinePrecision]), If[LessEqual[t$95$1, 5e-22], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+39], N[(y + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e221

   1. Initial program 75.9%

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 87.1%

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

   8. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 79.2

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

   10. Simplified 79.2%

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

   if -1e221 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999954e-22

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 84.0

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

   5. Simplified 84.0%

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

   if 4.99999999999999954e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000015e39

   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 95.0

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

   5. Simplified 95.0%

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

   if 5.00000000000000015e39 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 60.4

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

   5. Simplified 60.4%

      \[\leadsto x + \color{blue}{\frac{y \cdot t}{a}} \]
   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 66.0

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

   7. Applied egg-rr 66.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -1 \cdot 10^{+221}:\\ \;\;\;\;-\frac{t \cdot y}{z}\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{+39}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ t_2 := t \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+275}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))) (t_2 (* t (/ y a))))
   (if (<= t_1 -1e+275)
     t_2
     (if (<= t_1 5e-22) x (if (<= t_1 1e+63) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double t_2 = t * (y / a);
	double tmp;
	if (t_1 <= -1e+275) {
		tmp = t_2;
	} else if (t_1 <= 5e-22) {
		tmp = x;
	} else if (t_1 <= 1e+63) {
		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 = (t - z) / (a - z)
    t_2 = t * (y / a)
    if (t_1 <= (-1d+275)) then
        tmp = t_2
    else if (t_1 <= 5d-22) then
        tmp = x
    else if (t_1 <= 1d+63) 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 = (t - z) / (a - z);
	double t_2 = t * (y / a);
	double tmp;
	if (t_1 <= -1e+275) {
		tmp = t_2;
	} else if (t_1 <= 5e-22) {
		tmp = x;
	} else if (t_1 <= 1e+63) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (t - z) / (a - z)
	t_2 = t * (y / a)
	tmp = 0
	if t_1 <= -1e+275:
		tmp = t_2
	elif t_1 <= 5e-22:
		tmp = x
	elif t_1 <= 1e+63:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	t_2 = Float64(t * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -1e+275)
		tmp = t_2;
	elseif (t_1 <= 5e-22)
		tmp = x;
	elseif (t_1 <= 1e+63)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (t - z) / (a - z);
	t_2 = t * (y / a);
	tmp = 0.0;
	if (t_1 <= -1e+275)
		tmp = t_2;
	elseif (t_1 <= 5e-22)
		tmp = x;
	elseif (t_1 <= 1e+63)
		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[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+275], t$95$2, If[LessEqual[t$95$1, 5e-22], x, If[LessEqual[t$95$1, 1e+63], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -9.9999999999999996e274 or 1.00000000000000006e63 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 84.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 53.7

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

   5. Simplified 53.7%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 50.7

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

   7. Applied egg-rr 50.7%

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

   8. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

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

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

      3. /-lowering-/.f64 43.1

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

   10. Simplified 43.1%

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

   if -9.9999999999999996e274 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999954e-22

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 71.0%

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

   if 4.99999999999999954e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000006e63

   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 92.5

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

   5. Simplified 92.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -1 \cdot 10^{+275}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 10^{+63}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))))
   (if (<= t_1 -2e+65)
     (fma (/ y z) (- t) x)
     (if (<= t_1 4e-17) (fma y (/ (- t z) a) x) (fma y (- 1.0 (/ t z)) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double tmp;
	if (t_1 <= -2e+65) {
		tmp = fma((y / z), -t, x);
	} else if (t_1 <= 4e-17) {
		tmp = fma(y, ((t - z) / a), x);
	} else {
		tmp = fma(y, (1.0 - (t / z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -2e+65)
		tmp = fma(Float64(y / z), Float64(-t), x);
	elseif (t_1 <= 4e-17)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	else
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+65], N[(N[(y / z), $MachinePrecision] * (-t) + x), $MachinePrecision], If[LessEqual[t$95$1, 4e-17], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e65

   1. Initial program 91.5%

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 72.8%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 72.8

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

   10. Simplified 72.8%

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

   if -2e65 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000029e-17

   1. Initial program 99.4%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.0

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

   5. Simplified 99.0%

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

   if 4.00000000000000029e-17 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 86.9

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

   5. Simplified 86.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -t, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+39}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))))
   (if (<= t_1 5e-22)
     (fma y (/ t a) x)
     (if (<= t_1 5e+39) (+ y x) (fma t (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double tmp;
	if (t_1 <= 5e-22) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 5e+39) {
		tmp = y + x;
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= 5e-22)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 5e+39)
		tmp = Float64(y + x);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-22], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+39], N[(y + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999954e-22

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 79.6

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

   5. Simplified 79.6%

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

   if 4.99999999999999954e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000015e39

   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 95.0

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

   5. Simplified 95.0%

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

   if 5.00000000000000015e39 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 91.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 60.4

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

   5. Simplified 60.4%

      \[\leadsto x + \color{blue}{\frac{y \cdot t}{a}} \]
   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 66.0

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

   7. Applied egg-rr 66.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{+39}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a - z}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+39}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- a z))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 5e-22) t_2 (if (<= t_1 5e+39) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (a - z);
	double t_2 = fma(t, (y / a), x);
	double tmp;
	if (t_1 <= 5e-22) {
		tmp = t_2;
	} else if (t_1 <= 5e+39) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(a - z))
	t_2 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 5e-22)
		tmp = t_2;
	elseif (t_1 <= 5e+39)
		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[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-22], t$95$2, If[LessEqual[t$95$1, 5e+39], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 4.99999999999999954e-22 or 5.00000000000000015e39 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 96.1%

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

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

   5. Simplified 70.6%

      \[\leadsto x + \color{blue}{\frac{y \cdot t}{a}} \]
   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 76.3

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

   7. Applied egg-rr 76.3%

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

   if 4.99999999999999954e-22 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000015e39

   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 95.0

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

   5. Simplified 95.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{t - z}{a - z} \leq 5 \cdot 10^{+39}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq 2.4 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- t z) (- a z)) 2.4e-20) x (+ y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((t - z) / (a - z)) <= 2.4e-20) {
		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 (((t - z) / (a - z)) <= 2.4d-20) 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 (((t - z) / (a - z)) <= 2.4e-20) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((t - z) / (a - z)) <= 2.4e-20:
		tmp = x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(t - z) / Float64(a - z)) <= 2.4e-20)
		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 (((t - z) / (a - z)) <= 2.4e-20)
		tmp = x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], 2.4e-20], x, N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.39999999999999993e-20

   1. Initial program 97.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 65.8%

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

   if 2.39999999999999993e-20 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 76.9

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

   5. Simplified 76.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - z}{a - z} \leq 2.4 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.5%

   \[x + y \cdot \frac{z - t}{z - a} \]
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 97.5

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

4. Applied egg-rr 97.5%

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

5. Final simplification 97.5%

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

Alternative 16: 52.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= y 3.4e+105) x y))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 3.4e+105) {
		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) :: tmp
    if (y <= 3.4d+105) 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 tmp;
	if (y <= 3.4e+105) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= 3.4e+105:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 3.4e+105)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= 3.4e+105)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, 3.4e+105], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 3.3999999999999999e105

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 62.2%

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

   if 3.3999999999999999e105 < y

   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 42.7

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

   5. Simplified 42.7%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 39.5%

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

Alternative 17: 51.3% 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 97.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 56.2%

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

Developer Target 1: 98.4% 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 2024198 
(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)))))