Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Percentage Accurate: 68.6% → 88.8%

Time: 13.8s

Alternatives: 23

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z - a, \frac{1}{t} \cdot \left(x - y\right), y\right)\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- z a) (* (/ 1.0 t) (- x y)) y)))
   (if (<= t -2.3e+141)
     t_1
     (if (<= t 4.7e+120) (fma (/ (- z t) (- a t)) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z - a), ((1.0 / t) * (x - y)), y);
	double tmp;
	if (t <= -2.3e+141) {
		tmp = t_1;
	} else if (t <= 4.7e+120) {
		tmp = fma(((z - t) / (a - t)), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z - a), Float64(Float64(1.0 / t) * Float64(x - y)), y)
	tmp = 0.0
	if (t <= -2.3e+141)
		tmp = t_1;
	elseif (t <= 4.7e+120)
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - a), $MachinePrecision] * N[(N[(1.0 / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.3e+141], t$95$1, If[LessEqual[t, 4.7e+120], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.3000000000000002e141 or 4.69999999999999993e120 < t

   1. Initial program 37.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 84.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 87.2

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

   7. Applied egg-rr 87.2%

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

   if -2.3000000000000002e141 < t < 4.69999999999999993e120

   1. Initial program 83.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 92.2

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

   4. Applied egg-rr 92.2%

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

Alternative 2: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -2.3e+142)
     t_1
     (if (<= t -6e-12)
       (* y (/ (- z t) (- a t)))
       (if (<= t 1.42e-21) (fma (/ z (- a t)) (- y x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -2.3e+142) {
		tmp = t_1;
	} else if (t <= -6e-12) {
		tmp = y * ((z - t) / (a - t));
	} else if (t <= 1.42e-21) {
		tmp = fma((z / (a - t)), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -2.3e+142)
		tmp = t_1;
	elseif (t <= -6e-12)
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	elseif (t <= 1.42e-21)
		tmp = fma(Float64(z / Float64(a - t)), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.3e+142], t$95$1, If[LessEqual[t, -6e-12], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.42e-21], N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.30000000000000002e142 or 1.42e-21 < t

   1. Initial program 46.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 82.4%

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

   if -2.30000000000000002e142 < t < -6.0000000000000003e-12

   1. Initial program 63.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 55.0

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

   5. Simplified 55.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 71.5

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

   7. Applied egg-rr 71.5%

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

   if -6.0000000000000003e-12 < t < 1.42e-21

   1. Initial program 90.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 93.6

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

   4. Applied egg-rr 93.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 83.5

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

   7. Simplified 83.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -2.3e+142)
     t_1
     (if (<= t -6e-12)
       (* y (/ (- z t) (- a t)))
       (if (<= t 7.5e-26) (fma (- z t) (/ (- y x) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -2.3e+142) {
		tmp = t_1;
	} else if (t <= -6e-12) {
		tmp = y * ((z - t) / (a - t));
	} else if (t <= 7.5e-26) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -2.3e+142)
		tmp = t_1;
	elseif (t <= -6e-12)
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	elseif (t <= 7.5e-26)
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.3e+142], t$95$1, If[LessEqual[t, -6e-12], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e-26], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.30000000000000002e142 or 7.4999999999999994e-26 < t

   1. Initial program 46.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 82.4%

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

   if -2.30000000000000002e142 < t < -6.0000000000000003e-12

   1. Initial program 63.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 55.0

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

   5. Simplified 55.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 71.5

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

   7. Applied egg-rr 71.5%

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

   if -6.0000000000000003e-12 < t < 7.4999999999999994e-26

   1. Initial program 90.3%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 75.7

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

   5. Simplified 75.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -2.3e+142)
     t_1
     (if (<= t -2.6e-12)
       (* y (/ (- z t) (- a t)))
       (if (<= t 7.5e-26) (fma (/ z a) (- y x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -2.3e+142) {
		tmp = t_1;
	} else if (t <= -2.6e-12) {
		tmp = y * ((z - t) / (a - t));
	} else if (t <= 7.5e-26) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -2.3e+142)
		tmp = t_1;
	elseif (t <= -2.6e-12)
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	elseif (t <= 7.5e-26)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.3e+142], t$95$1, If[LessEqual[t, -2.6e-12], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e-26], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.30000000000000002e142 or 7.4999999999999994e-26 < t

   1. Initial program 46.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 82.4%

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

   if -2.30000000000000002e142 < t < -2.59999999999999983e-12

   1. Initial program 63.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 55.0

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

   5. Simplified 55.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 71.5

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

   7. Applied egg-rr 71.5%

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

   if -2.59999999999999983e-12 < t < 7.4999999999999994e-26

   1. Initial program 90.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 93.6

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

   4. Applied egg-rr 93.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 73.4

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

   7. Simplified 73.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 1.96 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- x y) t) y)))
   (if (<= t -1.9e+141)
     t_1
     (if (<= t -2.6e-12)
       (* y (/ (- z t) (- a t)))
       (if (<= t 1.96e-25) (fma (/ z a) (- y x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((x - y) / t), y);
	double tmp;
	if (t <= -1.9e+141) {
		tmp = t_1;
	} else if (t <= -2.6e-12) {
		tmp = y * ((z - t) / (a - t));
	} else if (t <= 1.96e-25) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(x - y) / t), y)
	tmp = 0.0
	if (t <= -1.9e+141)
		tmp = t_1;
	elseif (t <= -2.6e-12)
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	elseif (t <= 1.96e-25)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.9e+141], t$95$1, If[LessEqual[t, -2.6e-12], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.96e-25], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.89999999999999988e141 or 1.96e-25 < t

   1. Initial program 45.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 81.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. lower-fma.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 69.9

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

   8. Simplified 69.9%

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

   if -1.89999999999999988e141 < t < -2.59999999999999983e-12

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 56.3

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

   5. Simplified 56.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 73.2

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

   7. Applied egg-rr 73.2%

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

   if -2.59999999999999983e-12 < t < 1.96e-25

   1. Initial program 90.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 93.6

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

   4. Applied egg-rr 93.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 73.4

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

   7. Simplified 73.4%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 1.96 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{elif}\;t \leq 1.96 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- x y) t) y)))
   (if (<= t -1.85e+141)
     t_1
     (if (<= t -2.6e-12)
       (* (- z t) (/ y (- a t)))
       (if (<= t 1.96e-25) (fma (/ z a) (- y x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((x - y) / t), y);
	double tmp;
	if (t <= -1.85e+141) {
		tmp = t_1;
	} else if (t <= -2.6e-12) {
		tmp = (z - t) * (y / (a - t));
	} else if (t <= 1.96e-25) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(x - y) / t), y)
	tmp = 0.0
	if (t <= -1.85e+141)
		tmp = t_1;
	elseif (t <= -2.6e-12)
		tmp = Float64(Float64(z - t) * Float64(y / Float64(a - t)));
	elseif (t <= 1.96e-25)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.85e+141], t$95$1, If[LessEqual[t, -2.6e-12], N[(N[(z - t), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.96e-25], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.8500000000000001e141 or 1.96e-25 < t

   1. Initial program 45.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 81.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. lower-fma.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 69.9

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

   8. Simplified 69.9%

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

   if -1.8500000000000001e141 < t < -2.59999999999999983e-12

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 56.3

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

   5. Simplified 56.3%

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

      1. lift--.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.1

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

   7. Applied egg-rr 73.1%

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

   if -2.59999999999999983e-12 < t < 1.96e-25

   1. Initial program 90.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 93.6

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

   4. Applied egg-rr 93.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 73.4

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

   7. Simplified 73.4%

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

Alternative 7: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -2.8e+150)
     t_1
     (if (<= t 4.7e+120) (fma (/ (- z t) (- a t)) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -2.8e+150) {
		tmp = t_1;
	} else if (t <= 4.7e+120) {
		tmp = fma(((z - t) / (a - t)), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -2.8e+150)
		tmp = t_1;
	elseif (t <= 4.7e+120)
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.8e+150], t$95$1, If[LessEqual[t, 4.7e+120], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.80000000000000009e150 or 4.69999999999999993e120 < t

   1. Initial program 37.5%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 84.9%

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

   if -2.80000000000000009e150 < t < 4.69999999999999993e120

   1. Initial program 82.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 91.9

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

   4. Applied egg-rr 91.9%

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

Alternative 8: 43.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-250}:\\ \;\;\;\;\frac{\left(z - a\right) \cdot x}{t}\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- 1.0 (/ z t)))))
   (if (<= y -5.7e-72)
     t_1
     (if (<= y 1.95e-250)
       (/ (* (- z a) x) t)
       (if (<= y 155000000.0) (fma (- x) (/ z a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (1.0 - (z / t));
	double tmp;
	if (y <= -5.7e-72) {
		tmp = t_1;
	} else if (y <= 1.95e-250) {
		tmp = ((z - a) * x) / t;
	} else if (y <= 155000000.0) {
		tmp = fma(-x, (z / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(1.0 - Float64(z / t)))
	tmp = 0.0
	if (y <= -5.7e-72)
		tmp = t_1;
	elseif (y <= 1.95e-250)
		tmp = Float64(Float64(Float64(z - a) * x) / t);
	elseif (y <= 155000000.0)
		tmp = fma(Float64(-x), Float64(z / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.7e-72], t$95$1, If[LessEqual[y, 1.95e-250], N[(N[(N[(z - a), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[y, 155000000.0], N[((-x) * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.7000000000000003e-72 or 1.55e8 < y

   1. Initial program 66.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 56.7

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

   5. Simplified 56.7%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 52.2

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

   8. Simplified 52.2%

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

   if -5.7000000000000003e-72 < y < 1.95000000000000014e-250

   1. Initial program 70.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 57.5%

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

   6. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 51.8

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

   8. Simplified 51.8%

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

   if 1.95000000000000014e-250 < y < 1.55e8

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 68.4

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

   5. Simplified 68.4%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 60.8

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

   8. Simplified 60.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 64.3

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

   10. Applied egg-rr 64.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{-72}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-250}:\\ \;\;\;\;\frac{\left(z - a\right) \cdot x}{t}\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left(z - a\right) \cdot x}{t}\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;x - \frac{z \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- 1.0 (/ z t)))))
   (if (<= y -5.7e-72)
     t_1
     (if (<= y 3.5e-178)
       (/ (* (- z a) x) t)
       (if (<= y 155000000.0) (- x (/ (* z x) a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (1.0 - (z / t));
	double tmp;
	if (y <= -5.7e-72) {
		tmp = t_1;
	} else if (y <= 3.5e-178) {
		tmp = ((z - a) * x) / t;
	} else if (y <= 155000000.0) {
		tmp = x - ((z * x) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (1.0d0 - (z / t))
    if (y <= (-5.7d-72)) then
        tmp = t_1
    else if (y <= 3.5d-178) then
        tmp = ((z - a) * x) / t
    else if (y <= 155000000.0d0) then
        tmp = x - ((z * x) / a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (1.0 - (z / t));
	double tmp;
	if (y <= -5.7e-72) {
		tmp = t_1;
	} else if (y <= 3.5e-178) {
		tmp = ((z - a) * x) / t;
	} else if (y <= 155000000.0) {
		tmp = x - ((z * x) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (1.0 - (z / t))
	tmp = 0
	if y <= -5.7e-72:
		tmp = t_1
	elif y <= 3.5e-178:
		tmp = ((z - a) * x) / t
	elif y <= 155000000.0:
		tmp = x - ((z * x) / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(1.0 - Float64(z / t)))
	tmp = 0.0
	if (y <= -5.7e-72)
		tmp = t_1;
	elseif (y <= 3.5e-178)
		tmp = Float64(Float64(Float64(z - a) * x) / t);
	elseif (y <= 155000000.0)
		tmp = Float64(x - Float64(Float64(z * x) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (1.0 - (z / t));
	tmp = 0.0;
	if (y <= -5.7e-72)
		tmp = t_1;
	elseif (y <= 3.5e-178)
		tmp = ((z - a) * x) / t;
	elseif (y <= 155000000.0)
		tmp = x - ((z * x) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.7e-72], t$95$1, If[LessEqual[y, 3.5e-178], N[(N[(N[(z - a), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[y, 155000000.0], N[(x - N[(N[(z * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.7000000000000003e-72 or 1.55e8 < y

   1. Initial program 66.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 56.7

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

   5. Simplified 56.7%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 52.2

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

   8. Simplified 52.2%

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

   if -5.7000000000000003e-72 < y < 3.49999999999999983e-178

   1. Initial program 68.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 53.0%

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

   6. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 49.8

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

   8. Simplified 49.8%

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

   if 3.49999999999999983e-178 < y < 1.55e8

   1. Initial program 86.3%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 67.9

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

   8. Simplified 67.9%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{-72}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left(z - a\right) \cdot x}{t}\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;x - \frac{z \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{z \cdot x}{a}\\ \mathbf{if}\;a \leq -3.1 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;z \cdot \frac{y - x}{a}\\ \mathbf{elif}\;a \leq 13500000:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* z x) a))))
   (if (<= a -3.1e+95)
     t_1
     (if (<= a -9.2e-8)
       (* z (/ (- y x) a))
       (if (<= a 13500000.0) (* y (- 1.0 (/ z t))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((z * x) / a);
	double tmp;
	if (a <= -3.1e+95) {
		tmp = t_1;
	} else if (a <= -9.2e-8) {
		tmp = z * ((y - x) / a);
	} else if (a <= 13500000.0) {
		tmp = y * (1.0 - (z / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - ((z * x) / a)
    if (a <= (-3.1d+95)) then
        tmp = t_1
    else if (a <= (-9.2d-8)) then
        tmp = z * ((y - x) / a)
    else if (a <= 13500000.0d0) then
        tmp = y * (1.0d0 - (z / t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((z * x) / a);
	double tmp;
	if (a <= -3.1e+95) {
		tmp = t_1;
	} else if (a <= -9.2e-8) {
		tmp = z * ((y - x) / a);
	} else if (a <= 13500000.0) {
		tmp = y * (1.0 - (z / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - ((z * x) / a)
	tmp = 0
	if a <= -3.1e+95:
		tmp = t_1
	elif a <= -9.2e-8:
		tmp = z * ((y - x) / a)
	elif a <= 13500000.0:
		tmp = y * (1.0 - (z / t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(z * x) / a))
	tmp = 0.0
	if (a <= -3.1e+95)
		tmp = t_1;
	elseif (a <= -9.2e-8)
		tmp = Float64(z * Float64(Float64(y - x) / a));
	elseif (a <= 13500000.0)
		tmp = Float64(y * Float64(1.0 - Float64(z / t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - ((z * x) / a);
	tmp = 0.0;
	if (a <= -3.1e+95)
		tmp = t_1;
	elseif (a <= -9.2e-8)
		tmp = z * ((y - x) / a);
	elseif (a <= 13500000.0)
		tmp = y * (1.0 - (z / t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(z * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.1e+95], t$95$1, If[LessEqual[a, -9.2e-8], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 13500000.0], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.1000000000000003e95 or 1.35e7 < a

   1. Initial program 68.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 52.0

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

   5. Simplified 52.0%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 47.4

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

   8. Simplified 47.4%

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

   if -3.1000000000000003e95 < a < -9.2000000000000003e-8

   1. Initial program 71.8%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 57.0

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

   5. Simplified 57.0%

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

   6. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 48.9

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

   8. Simplified 48.9%

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

   if -9.2000000000000003e-8 < a < 1.35e7

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 55.2

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

   5. Simplified 55.2%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 56.8

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

   8. Simplified 56.8%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+95}:\\ \;\;\;\;x - \frac{z \cdot x}{a}\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;z \cdot \frac{y - x}{a}\\ \mathbf{elif}\;a \leq 13500000:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-55}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-221}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{a}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -9.2e-55)
   y
   (if (<= t 4.6e-221)
     (/ (* z y) a)
     (if (<= t 9.5e-22) (fma x (/ z t) x) (fma y (/ a t) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9.2e-55) {
		tmp = y;
	} else if (t <= 4.6e-221) {
		tmp = (z * y) / a;
	} else if (t <= 9.5e-22) {
		tmp = fma(x, (z / t), x);
	} else {
		tmp = fma(y, (a / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -9.2e-55)
		tmp = y;
	elseif (t <= 4.6e-221)
		tmp = Float64(Float64(z * y) / a);
	elseif (t <= 9.5e-22)
		tmp = fma(x, Float64(z / t), x);
	else
		tmp = fma(y, Float64(a / t), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -9.2e-55], y, If[LessEqual[t, 4.6e-221], N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 9.5e-22], N[(x * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(a / t), $MachinePrecision] + y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -9.20000000000000046e-55

   1. Initial program 55.9%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 32.2

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

   5. Simplified 32.2%

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

      1. lift--.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(x - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \left(x - x\right)} \]

      6. lower--.f64 38.6

         \[\leadsto y - \color{blue}{\left(x - x\right)} \]

   7. Applied egg-rr 38.6%

      \[\leadsto \color{blue}{y - \left(x - x\right)} \]
   8. Step-by-step derivation

      1. +-inverses N/A

         \[\leadsto y - \color{blue}{0} \]

      2. --rgt-identity 38.6

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

   9. Applied egg-rr 38.6%

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

   if -9.20000000000000046e-55 < t < 4.6e-221

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 52.3

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

   5. Simplified 52.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 39.6

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

   8. Simplified 39.6%

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

   if 4.6e-221 < t < 9.4999999999999994e-22

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 60.6

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

   5. Simplified 60.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 15.4

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

   8. Simplified 15.4%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 34.1

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

   11. Simplified 34.1%

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

   if 9.4999999999999994e-22 < t

   1. Initial program 52.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 43.9

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

   5. Simplified 43.9%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 48.0

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

   8. Simplified 48.0%

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

   9. Taylor expanded in t around inf

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. mul-1-neg N/A

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

       5. remove-double-neg N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-/.f64 47.6

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

   11. Simplified 47.6%

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

Alternative 12: 35.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-55}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-221}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -9.2e-55)
   y
   (if (<= t 4.6e-221) (/ (* z y) a) (if (<= t 9.5e-22) (fma x (/ z t) x) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9.2e-55) {
		tmp = y;
	} else if (t <= 4.6e-221) {
		tmp = (z * y) / a;
	} else if (t <= 9.5e-22) {
		tmp = fma(x, (z / t), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -9.2e-55)
		tmp = y;
	elseif (t <= 4.6e-221)
		tmp = Float64(Float64(z * y) / a);
	elseif (t <= 9.5e-22)
		tmp = fma(x, Float64(z / t), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -9.2e-55], y, If[LessEqual[t, 4.6e-221], N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 9.5e-22], N[(x * N[(z / t), $MachinePrecision] + x), $MachinePrecision], y]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.20000000000000046e-55 or 9.4999999999999994e-22 < t

   1. Initial program 54.6%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 34.5

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

   5. Simplified 34.5%

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

      1. lift--.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(x - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \left(x - x\right)} \]

      6. lower--.f64 42.6

         \[\leadsto y - \color{blue}{\left(x - x\right)} \]

   7. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{y - \left(x - x\right)} \]
   8. Step-by-step derivation

      1. +-inverses N/A

         \[\leadsto y - \color{blue}{0} \]

      2. --rgt-identity 42.6

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

   9. Applied egg-rr 42.6%

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

   if -9.20000000000000046e-55 < t < 4.6e-221

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 52.3

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

   5. Simplified 52.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 39.6

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

   8. Simplified 39.6%

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

   if 4.6e-221 < t < 9.4999999999999994e-22

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 60.6

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

   5. Simplified 60.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 15.4

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

   8. Simplified 15.4%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 34.1

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

   11. Simplified 34.1%

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

Alternative 13: 62.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-176}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= y -1.15e-176)
     t_1
     (if (<= y 155000000.0) (fma (/ z (- a t)) (- x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (y <= -1.15e-176) {
		tmp = t_1;
	} else if (y <= 155000000.0) {
		tmp = fma((z / (a - t)), -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (y <= -1.15e-176)
		tmp = t_1;
	elseif (y <= 155000000.0)
		tmp = fma(Float64(z / Float64(a - t)), Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e-176], t$95$1, If[LessEqual[y, 155000000.0], N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1500000000000001e-176 or 1.55e8 < y

   1. Initial program 67.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 55.5

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

   5. Simplified 55.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 73.6

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

   7. Applied egg-rr 73.6%

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

   if -1.1500000000000001e-176 < y < 1.55e8

   1. Initial program 75.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 76.5

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

   4. Applied egg-rr 76.5%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 70.5

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

   7. Simplified 70.5%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.5

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

   10. Simplified 64.5%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-176}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z a) (- y x) x)))
   (if (<= a -2.2e-8) t_1 (if (<= a 2.6) (fma z (/ (- x y) t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / a), (y - x), x);
	double tmp;
	if (a <= -2.2e-8) {
		tmp = t_1;
	} else if (a <= 2.6) {
		tmp = fma(z, ((x - y) / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / a), Float64(y - x), x)
	tmp = 0.0
	if (a <= -2.2e-8)
		tmp = t_1;
	elseif (a <= 2.6)
		tmp = fma(z, Float64(Float64(x - y) / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.2e-8], t$95$1, If[LessEqual[a, 2.6], N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.1999999999999998e-8 or 2.60000000000000009 < a

   1. Initial program 69.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      9. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      14. lower-/.f64 89.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{a - t}}, y - x, x\right) \]

   4. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 65.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]

   7. Simplified 65.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]

   if -2.1999999999999998e-8 < a < 2.60000000000000009

   1. Initial program 69.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Simplified 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{y + \frac{z \cdot \left(x - y\right)}{t}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(x - y\right)}{t} + y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{x - y}{t}} + y \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} + y \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{x}{t} - \frac{y}{t}, y\right)} \]

      5. div-sub N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x - y}{t}}, y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x - y}{t}}, y\right) \]

      7. lower--.f64 75.1

         \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{x - y}}{t}, y\right) \]

   8. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.7:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- y x) a) x)))
   (if (<= a -2.2e-8) t_1 (if (<= a 2.7) (fma z (/ (- x y) t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((y - x) / a), x);
	double tmp;
	if (a <= -2.2e-8) {
		tmp = t_1;
	} else if (a <= 2.7) {
		tmp = fma(z, ((x - y) / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(y - x) / a), x)
	tmp = 0.0
	if (a <= -2.2e-8)
		tmp = t_1;
	elseif (a <= 2.7)
		tmp = fma(z, Float64(Float64(x - y) / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.2e-8], t$95$1, If[LessEqual[a, 2.7], N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.1999999999999998e-8 or 2.7000000000000002 < a

   1. Initial program 69.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{y - x}{a}}, x\right) \]

      5. lower--.f64 64.6

         \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{y - x}}{a}, x\right) \]

   5. Simplified 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)} \]

   if -2.1999999999999998e-8 < a < 2.7000000000000002

   1. Initial program 69.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Simplified 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{y + \frac{z \cdot \left(x - y\right)}{t}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(x - y\right)}{t} + y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{x - y}{t}} + y \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} + y \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{x}{t} - \frac{y}{t}, y\right)} \]

      5. div-sub N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x - y}{t}}, y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x - y}{t}}, y\right) \]

      7. lower--.f64 75.1

         \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{x - y}}{t}, y\right) \]

   8. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+103}:\\ \;\;\;\;x - \frac{z \cdot x}{a}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -7.2e+103)
   (- x (/ (* z x) a))
   (if (<= a 2.1e+139) (fma z (/ (- x y) t) y) (fma (- x) (/ z a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7.2e+103) {
		tmp = x - ((z * x) / a);
	} else if (a <= 2.1e+139) {
		tmp = fma(z, ((x - y) / t), y);
	} else {
		tmp = fma(-x, (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -7.2e+103)
		tmp = Float64(x - Float64(Float64(z * x) / a));
	elseif (a <= 2.1e+139)
		tmp = fma(z, Float64(Float64(x - y) / t), y);
	else
		tmp = fma(Float64(-x), Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -7.2e+103], N[(x - N[(N[(z * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.1e+139], N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], N[((-x) * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.20000000000000033e103

   1. Initial program 71.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z - t}{a - t}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(z - t\right)}{a - t}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - t\right) \cdot x}}{a - t}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - t\right) \cdot \frac{x}{a - t}}\right)\right) + 1 \cdot x \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right)} + 1 \cdot x \]

      10. *-lft-identity N/A

          \[\leadsto \left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right) + \color{blue}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}\right)}, x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}\right)}, x\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t} + \left(\mathsf{neg}\left(a\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      20. lower-neg.f64 66.0

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{t + \color{blue}{\left(-a\right)}}, x\right) \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{x}{t + \left(-a\right)}, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot z}{a}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot z}{a}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{a}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{x \cdot z}{a}} \]

      5. lower-*.f64 64.0

         \[\leadsto x - \frac{\color{blue}{x \cdot z}}{a} \]

   8. Simplified 64.0%

      \[\leadsto \color{blue}{x - \frac{x \cdot z}{a}} \]

   if -7.20000000000000033e103 < a < 2.0999999999999999e139

   1. Initial program 68.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Simplified 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{y + \frac{z \cdot \left(x - y\right)}{t}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(x - y\right)}{t} + y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{x - y}{t}} + y \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} + y \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{x}{t} - \frac{y}{t}, y\right)} \]

      5. div-sub N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x - y}{t}}, y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x - y}{t}}, y\right) \]

      7. lower--.f64 63.0

         \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{x - y}}{t}, y\right) \]

   8. Simplified 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)} \]

   if 2.0999999999999999e139 < a

   1. Initial program 77.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z - t}{a - t}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(z - t\right)}{a - t}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - t\right) \cdot x}}{a - t}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - t\right) \cdot \frac{x}{a - t}}\right)\right) + 1 \cdot x \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right)} + 1 \cdot x \]

      10. *-lft-identity N/A

          \[\leadsto \left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right) + \color{blue}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}\right)}, x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}\right)}, x\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t} + \left(\mathsf{neg}\left(a\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      20. lower-neg.f64 53.9

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{t + \color{blue}{\left(-a\right)}}, x\right) \]

   5. Simplified 53.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{x}{t + \left(-a\right)}, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot z}{a}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot z}{a}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{a}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{x \cdot z}{a}} \]

      5. lower-*.f64 49.3

         \[\leadsto x - \frac{\color{blue}{x \cdot z}}{a} \]

   8. Simplified 49.3%

      \[\leadsto \color{blue}{x - \frac{x \cdot z}{a}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{x \cdot z}}{a} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{x \cdot z}{a}} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{x \cdot z}{a}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot z}{a}\right)\right) + x} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot z}{a}}\right)\right) + x \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot z}}{a}\right)\right) + x \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z}{a}}\right)\right) + x \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{z}{a}} + x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{z}{a}, x\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, \frac{z}{a}, x\right) \]

      11. lower-/.f64 53.9

          \[\leadsto \mathsf{fma}\left(-x, \color{blue}{\frac{z}{a}}, x\right) \]

   10. Applied egg-rr 53.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{z}{a}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+103}:\\ \;\;\;\;x - \frac{z \cdot x}{a}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-36}:\\ \;\;\;\;z \cdot \frac{y - x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- 1.0 (/ z t)))))
   (if (<= t -7.2e-42) t_1 (if (<= t 7.6e-36) (* z (/ (- y x) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (1.0 - (z / t));
	double tmp;
	if (t <= -7.2e-42) {
		tmp = t_1;
	} else if (t <= 7.6e-36) {
		tmp = z * ((y - x) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (1.0d0 - (z / t))
    if (t <= (-7.2d-42)) then
        tmp = t_1
    else if (t <= 7.6d-36) then
        tmp = z * ((y - x) / a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (1.0 - (z / t));
	double tmp;
	if (t <= -7.2e-42) {
		tmp = t_1;
	} else if (t <= 7.6e-36) {
		tmp = z * ((y - x) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (1.0 - (z / t))
	tmp = 0
	if t <= -7.2e-42:
		tmp = t_1
	elif t <= 7.6e-36:
		tmp = z * ((y - x) / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(1.0 - Float64(z / t)))
	tmp = 0.0
	if (t <= -7.2e-42)
		tmp = t_1;
	elseif (t <= 7.6e-36)
		tmp = Float64(z * Float64(Float64(y - x) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (1.0 - (z / t));
	tmp = 0.0;
	if (t <= -7.2e-42)
		tmp = t_1;
	elseif (t <= 7.6e-36)
		tmp = z * ((y - x) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.2e-42], t$95$1, If[LessEqual[t, 7.6e-36], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.2000000000000004e-42 or 7.59999999999999942e-36 < t

   1. Initial program 54.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{a - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(z - t\right)}}{a - t} \]

      4. lower--.f64 41.4

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

   5. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{t}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y \cdot \left(z - t\right)}{t}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot \frac{z - t}{t}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{z - t}{t}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{t}\right)} \]

      5. div-sub N/A

         \[\leadsto y \cdot \left(-1 \cdot \color{blue}{\left(\frac{z}{t} - \frac{t}{t}\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto y \cdot \left(-1 \cdot \color{blue}{\left(\frac{z}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)}\right) \]

      7. *-inverses N/A

         \[\leadsto y \cdot \left(-1 \cdot \left(\frac{z}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto y \cdot \left(-1 \cdot \left(\frac{z}{t} + \color{blue}{-1}\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{z}{t} + -1 \cdot -1\right)} \]

      10. metadata-eval N/A

          \[\leadsto y \cdot \left(-1 \cdot \frac{z}{t} + \color{blue}{1}\right) \]

      11. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(1 + -1 \cdot \frac{z}{t}\right)} \]

      13. mul-1-neg N/A

          \[\leadsto y \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{z}{t}\right)\right)}\right) \]

      14. unsub-neg N/A

          \[\leadsto y \cdot \color{blue}{\left(1 - \frac{z}{t}\right)} \]

      15. lower--.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(1 - \frac{z}{t}\right)} \]

      16. lower-/.f64 48.8

          \[\leadsto y \cdot \left(1 - \color{blue}{\frac{z}{t}}\right) \]

   8. Simplified 48.8%

      \[\leadsto \color{blue}{y \cdot \left(1 - \frac{z}{t}\right)} \]

   if -7.2000000000000004e-42 < t < 7.59999999999999942e-36

   1. Initial program 91.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{a - t} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{z \cdot \color{blue}{\left(y - x\right)}}{a - t} \]

      6. lower--.f64 54.8

         \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a - t}} \]

   5. Simplified 54.8%

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a}} \]

      4. lower--.f64 45.7

         \[\leadsto z \cdot \frac{\color{blue}{y - x}}{a} \]

   8. Simplified 45.7%

      \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 42.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-36}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- 1.0 (/ z t)))))
   (if (<= t -8.6e-55) t_1 (if (<= t 8e-36) (* y (/ z (- a t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (1.0 - (z / t));
	double tmp;
	if (t <= -8.6e-55) {
		tmp = t_1;
	} else if (t <= 8e-36) {
		tmp = y * (z / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (1.0d0 - (z / t))
    if (t <= (-8.6d-55)) then
        tmp = t_1
    else if (t <= 8d-36) then
        tmp = y * (z / (a - t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (1.0 - (z / t));
	double tmp;
	if (t <= -8.6e-55) {
		tmp = t_1;
	} else if (t <= 8e-36) {
		tmp = y * (z / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (1.0 - (z / t))
	tmp = 0
	if t <= -8.6e-55:
		tmp = t_1
	elif t <= 8e-36:
		tmp = y * (z / (a - t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(1.0 - Float64(z / t)))
	tmp = 0.0
	if (t <= -8.6e-55)
		tmp = t_1;
	elseif (t <= 8e-36)
		tmp = Float64(y * Float64(z / Float64(a - t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (1.0 - (z / t));
	tmp = 0.0;
	if (t <= -8.6e-55)
		tmp = t_1;
	elseif (t <= 8e-36)
		tmp = y * (z / (a - t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.6e-55], t$95$1, If[LessEqual[t, 8e-36], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.60000000000000021e-55 or 7.9999999999999995e-36 < t

   1. Initial program 55.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{a - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(z - t\right)}}{a - t} \]

      4. lower--.f64 41.0

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

   5. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{t}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y \cdot \left(z - t\right)}{t}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot \frac{z - t}{t}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{z - t}{t}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{t}\right)} \]

      5. div-sub N/A

         \[\leadsto y \cdot \left(-1 \cdot \color{blue}{\left(\frac{z}{t} - \frac{t}{t}\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto y \cdot \left(-1 \cdot \color{blue}{\left(\frac{z}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)}\right) \]

      7. *-inverses N/A

         \[\leadsto y \cdot \left(-1 \cdot \left(\frac{z}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto y \cdot \left(-1 \cdot \left(\frac{z}{t} + \color{blue}{-1}\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{z}{t} + -1 \cdot -1\right)} \]

      10. metadata-eval N/A

          \[\leadsto y \cdot \left(-1 \cdot \frac{z}{t} + \color{blue}{1}\right) \]

      11. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(1 + -1 \cdot \frac{z}{t}\right)} \]

      13. mul-1-neg N/A

          \[\leadsto y \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{z}{t}\right)\right)}\right) \]

      14. unsub-neg N/A

          \[\leadsto y \cdot \color{blue}{\left(1 - \frac{z}{t}\right)} \]

      15. lower--.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(1 - \frac{z}{t}\right)} \]

      16. lower-/.f64 48.2

          \[\leadsto y \cdot \left(1 - \color{blue}{\frac{z}{t}}\right) \]

   8. Simplified 48.2%

      \[\leadsto \color{blue}{y \cdot \left(1 - \frac{z}{t}\right)} \]

   if -8.60000000000000021e-55 < t < 7.9999999999999995e-36

   1. Initial program 91.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{a - t} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{z \cdot \color{blue}{\left(y - x\right)}}{a - t} \]

      6. lower--.f64 55.8

         \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a - t}} \]

   5. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{a - t}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{a - t}} \]

      4. lower--.f64 42.8

         \[\leadsto y \cdot \frac{z}{\color{blue}{a - t}} \]

   8. Simplified 42.8%

      \[\leadsto \color{blue}{y \cdot \frac{z}{a - t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 40.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- 1.0 (/ z t)))))
   (if (<= y -2.5e-72) t_1 (if (<= y 155000000.0) (fma x (/ z t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (1.0 - (z / t));
	double tmp;
	if (y <= -2.5e-72) {
		tmp = t_1;
	} else if (y <= 155000000.0) {
		tmp = fma(x, (z / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(1.0 - Float64(z / t)))
	tmp = 0.0
	if (y <= -2.5e-72)
		tmp = t_1;
	elseif (y <= 155000000.0)
		tmp = fma(x, Float64(z / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e-72], t$95$1, If[LessEqual[y, 155000000.0], N[(x * N[(z / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4999999999999998e-72 or 1.55e8 < y

   1. Initial program 66.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{a - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(z - t\right)}}{a - t} \]

      4. lower--.f64 56.7

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

   5. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{t}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y \cdot \left(z - t\right)}{t}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot \frac{z - t}{t}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{z - t}{t}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{t}\right)} \]

      5. div-sub N/A

         \[\leadsto y \cdot \left(-1 \cdot \color{blue}{\left(\frac{z}{t} - \frac{t}{t}\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto y \cdot \left(-1 \cdot \color{blue}{\left(\frac{z}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)}\right) \]

      7. *-inverses N/A

         \[\leadsto y \cdot \left(-1 \cdot \left(\frac{z}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto y \cdot \left(-1 \cdot \left(\frac{z}{t} + \color{blue}{-1}\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{z}{t} + -1 \cdot -1\right)} \]

      10. metadata-eval N/A

          \[\leadsto y \cdot \left(-1 \cdot \frac{z}{t} + \color{blue}{1}\right) \]

      11. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(1 + -1 \cdot \frac{z}{t}\right)} \]

      13. mul-1-neg N/A

          \[\leadsto y \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{z}{t}\right)\right)}\right) \]

      14. unsub-neg N/A

          \[\leadsto y \cdot \color{blue}{\left(1 - \frac{z}{t}\right)} \]

      15. lower--.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(1 - \frac{z}{t}\right)} \]

      16. lower-/.f64 52.2

          \[\leadsto y \cdot \left(1 - \color{blue}{\frac{z}{t}}\right) \]

   8. Simplified 52.2%

      \[\leadsto \color{blue}{y \cdot \left(1 - \frac{z}{t}\right)} \]

   if -2.4999999999999998e-72 < y < 1.55e8

   1. Initial program 74.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z - t}{a - t}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(z - t\right)}{a - t}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - t\right) \cdot x}}{a - t}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - t\right) \cdot \frac{x}{a - t}}\right)\right) + 1 \cdot x \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right)} + 1 \cdot x \]

      10. *-lft-identity N/A

          \[\leadsto \left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right) + \color{blue}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}\right)}, x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}\right)}, x\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t} + \left(\mathsf{neg}\left(a\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      20. lower-neg.f64 61.4

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{t + \color{blue}{\left(-a\right)}}, x\right) \]

   5. Simplified 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{x}{t + \left(-a\right)}, x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{x \cdot \left(z - t\right)}{t}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(z - t\right)}{t} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{z - t}{t}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z - t}{t}, x\right)} \]

      4. div-sub N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t} - \frac{t}{t}}, x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)}, x\right) \]

      6. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{z}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{z}{t} + \color{blue}{-1}, x\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t} + -1}, x\right) \]

      9. lower-/.f64 19.2

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t}} + -1, x\right) \]

   8. Simplified 19.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{t} + -1, x\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t}}, x\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 31.1

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t}}, x\right) \]

   11. Simplified 31.1%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 33.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-55}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -9.2e-55) y (if (<= t 1.6e-25) (/ (* z y) a) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9.2e-55) {
		tmp = y;
	} else if (t <= 1.6e-25) {
		tmp = (z * y) / a;
	} 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 (t <= (-9.2d-55)) then
        tmp = y
    else if (t <= 1.6d-25) then
        tmp = (z * y) / a
    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 (t <= -9.2e-55) {
		tmp = y;
	} else if (t <= 1.6e-25) {
		tmp = (z * y) / a;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -9.2e-55:
		tmp = y
	elif t <= 1.6e-25:
		tmp = (z * y) / a
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -9.2e-55)
		tmp = y;
	elseif (t <= 1.6e-25)
		tmp = Float64(Float64(z * y) / a);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -9.2e-55)
		tmp = y;
	elseif (t <= 1.6e-25)
		tmp = (z * y) / a;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -9.2e-55], y, If[LessEqual[t, 1.6e-25], N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -9.20000000000000046e-55 or 1.6000000000000001e-25 < t

   1. Initial program 54.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 34.5

         \[\leadsto x + \color{blue}{\left(y - x\right)} \]

   5. Simplified 34.5%

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x + \color{blue}{\left(y - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) + x} \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(y - x\right)} + x \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(x - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \left(x - x\right)} \]

      6. lower--.f64 42.6

         \[\leadsto y - \color{blue}{\left(x - x\right)} \]

   7. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{y - \left(x - x\right)} \]
   8. Step-by-step derivation

      1. +-inverses N/A

         \[\leadsto y - \color{blue}{0} \]

      2. --rgt-identity 42.6

         \[\leadsto \color{blue}{y} \]

   9. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{y} \]

   if -9.20000000000000046e-55 < t < 1.6000000000000001e-25

   1. Initial program 91.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{a - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(z - t\right)}}{a - t} \]

      4. lower--.f64 44.2

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

   5. Simplified 44.2%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{y \cdot z}{a}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot y}}{a} \]

      3. lower-*.f64 31.3

         \[\leadsto \frac{\color{blue}{z \cdot y}}{a} \]

   8. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{z \cdot y}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 28.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-72}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{z \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -2.2e-72) y (if (<= y 3.3e-52) (/ (* z x) t) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.2e-72) {
		tmp = y;
	} else if (y <= 3.3e-52) {
		tmp = (z * x) / t;
	} 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 <= (-2.2d-72)) then
        tmp = y
    else if (y <= 3.3d-52) then
        tmp = (z * x) / t
    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 <= -2.2e-72) {
		tmp = y;
	} else if (y <= 3.3e-52) {
		tmp = (z * x) / t;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -2.2e-72:
		tmp = y
	elif y <= 3.3e-52:
		tmp = (z * x) / t
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -2.2e-72)
		tmp = y;
	elseif (y <= 3.3e-52)
		tmp = Float64(Float64(z * x) / t);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -2.2e-72)
		tmp = y;
	elseif (y <= 3.3e-52)
		tmp = (z * x) / t;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -2.2e-72], y, If[LessEqual[y, 3.3e-52], N[(N[(z * x), $MachinePrecision] / t), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000002e-72 or 3.29999999999999995e-52 < y

   1. Initial program 68.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 32.1

         \[\leadsto x + \color{blue}{\left(y - x\right)} \]

   5. Simplified 32.1%

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x + \color{blue}{\left(y - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) + x} \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(y - x\right)} + x \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{y - \left(x - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{y - \left(x - x\right)} \]

      6. lower--.f64 38.7

         \[\leadsto y - \color{blue}{\left(x - x\right)} \]

   7. Applied egg-rr 38.7%

      \[\leadsto \color{blue}{y - \left(x - x\right)} \]
   8. Step-by-step derivation

      1. +-inverses N/A

         \[\leadsto y - \color{blue}{0} \]

      2. --rgt-identity 38.7

         \[\leadsto \color{blue}{y} \]

   9. Applied egg-rr 38.7%

      \[\leadsto \color{blue}{y} \]

   if -2.20000000000000002e-72 < y < 3.29999999999999995e-52

   1. Initial program 72.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z - t}{a - t}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(z - t\right)}{a - t}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - t\right) \cdot x}}{a - t}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - t\right) \cdot \frac{x}{a - t}}\right)\right) + 1 \cdot x \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right)} + 1 \cdot x \]

      10. *-lft-identity N/A

          \[\leadsto \left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right) + \color{blue}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}\right)}, x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}\right)}, x\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t} + \left(\mathsf{neg}\left(a\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      20. lower-neg.f64 60.9

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{t + \color{blue}{\left(-a\right)}}, x\right) \]

   5. Simplified 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{x}{t + \left(-a\right)}, x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{x \cdot \left(z - t\right)}{t}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(z - t\right)}{t} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{z - t}{t}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z - t}{t}, x\right)} \]

      4. div-sub N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t} - \frac{t}{t}}, x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)}, x\right) \]

      6. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{z}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{z}{t} + \color{blue}{-1}, x\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t} + -1}, x\right) \]

      9. lower-/.f64 21.1

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{t}} + -1, x\right) \]

   8. Simplified 21.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{t} + -1, x\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot z}{t}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot z}{t}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{z \cdot x}}{t} \]

       3. lower-*.f64 28.6

          \[\leadsto \frac{\color{blue}{z \cdot x}}{t} \]

   11. Simplified 28.6%

       \[\leadsto \color{blue}{\frac{z \cdot x}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 25.0% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 y)

C

double code(double x, double y, double z, double t, double a) {
	return y;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = y
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return y;
}

Python

def code(x, y, z, t, a):
	return y

Julia

function code(x, y, z, t, a)
	return y
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = y;
end

Wolfram

code[x_, y_, z_, t_, a_] := y

Derivation

1. Initial program 69.8%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto x + \color{blue}{\left(y - x\right)} \]
4. Step-by-step derivation

   1. lower--.f64 23.1

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]

5. Simplified 23.1%

   \[\leadsto x + \color{blue}{\left(y - x\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(y - x\right) + x} \]

   3. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(y - x\right)} + x \]

   4. associate-+l- N/A

      \[\leadsto \color{blue}{y - \left(x - x\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \color{blue}{y - \left(x - x\right)} \]

   6. lower--.f64 27.9

      \[\leadsto y - \color{blue}{\left(x - x\right)} \]

7. Applied egg-rr 27.9%

   \[\leadsto \color{blue}{y - \left(x - x\right)} \]
8. Step-by-step derivation

   1. +-inverses N/A

      \[\leadsto y - \color{blue}{0} \]

   2. --rgt-identity 27.9

      \[\leadsto \color{blue}{y} \]

9. Applied egg-rr 27.9%

   \[\leadsto \color{blue}{y} \]
10. Add Preprocessing

Alternative 23: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 69.8%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t} + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot x + 1 \cdot x} \]

   3. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t}\right)\right)} \cdot x + 1 \cdot x \]

   4. distribute-lft-neg-out N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t} \cdot x\right)\right)} + 1 \cdot x \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z - t}{a - t}}\right)\right) + 1 \cdot x \]

   6. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(z - t\right)}{a - t}}\right)\right) + 1 \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - t\right) \cdot x}}{a - t}\right)\right) + 1 \cdot x \]

   8. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - t\right) \cdot \frac{x}{a - t}}\right)\right) + 1 \cdot x \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right)} + 1 \cdot x \]

   10. *-lft-identity N/A

       \[\leadsto \left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right) + \color{blue}{x} \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right) \]

   13. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

   14. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

   15. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}\right)}, x\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}\right)}, x\right) \]

   17. distribute-neg-in N/A

       \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

   18. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t} + \left(\mathsf{neg}\left(a\right)\right)}, x\right) \]

   19. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

   20. lower-neg.f64 37.0

       \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{t + \color{blue}{\left(-a\right)}}, x\right) \]

5. Simplified 37.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{x}{t + \left(-a\right)}, x\right)} \]

6. Taylor expanded in t around inf

   \[\leadsto \color{blue}{x + -1 \cdot x} \]
7. Step-by-step derivation

   1. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot x} \]

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot x \]

   3. mul0-lft 2.6

      \[\leadsto \color{blue}{0} \]

8. Simplified 2.6%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Developer Target 1: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))