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

Percentage Accurate: 68.2% → 89.1%

Time: 12.0s

Alternatives: 17

Speedup: 0.8×

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: 68.2% 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: 89.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+145}:\\ \;\;\;\;y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+118}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{z - a}{\frac{t}{x - y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2e+145)
   (+ y (* (/ (- y x) t) (- a z)))
   (if (<= t 4.1e+118)
     (+ x (/ (- y x) (/ (- a t) (- z t))))
     (+ y (/ (- z a) (/ t (- x y)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2e+145) {
		tmp = y + (((y - x) / t) * (a - z));
	} else if (t <= 4.1e+118) {
		tmp = x + ((y - x) / ((a - t) / (z - t)));
	} else {
		tmp = y + ((z - a) / (t / (x - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-2d+145)) then
        tmp = y + (((y - x) / t) * (a - z))
    else if (t <= 4.1d+118) then
        tmp = x + ((y - x) / ((a - t) / (z - t)))
    else
        tmp = y + ((z - a) / (t / (x - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2e+145) {
		tmp = y + (((y - x) / t) * (a - z));
	} else if (t <= 4.1e+118) {
		tmp = x + ((y - x) / ((a - t) / (z - t)));
	} else {
		tmp = y + ((z - a) / (t / (x - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -2e+145:
		tmp = y + (((y - x) / t) * (a - z))
	elif t <= 4.1e+118:
		tmp = x + ((y - x) / ((a - t) / (z - t)))
	else:
		tmp = y + ((z - a) / (t / (x - y)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2e+145)
		tmp = Float64(y + Float64(Float64(Float64(y - x) / t) * Float64(a - z)));
	elseif (t <= 4.1e+118)
		tmp = Float64(x + Float64(Float64(y - x) / Float64(Float64(a - t) / Float64(z - t))));
	else
		tmp = Float64(y + Float64(Float64(z - a) / Float64(t / Float64(x - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -2e+145)
		tmp = y + (((y - x) / t) * (a - z));
	elseif (t <= 4.1e+118)
		tmp = x + ((y - x) / ((a - t) / (z - t)));
	else
		tmp = y + ((z - a) / (t / (x - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2e+145], N[(y + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.1e+118], N[(x + N[(N[(y - x), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(z - a), $MachinePrecision] / N[(t / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2e145

   1. Initial program 15.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 93.0

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

   7. Applied rewrites 93.0%

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

   if -2e145 < t < 4.0999999999999997e118

   1. Initial program 81.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if 4.0999999999999997e118 < t

   1. Initial program 29.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 90.8

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

   7. Applied rewrites 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 90.9

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

   9. Applied rewrites 90.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+145}:\\ \;\;\;\;y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+118}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{z - a}{\frac{t}{x - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ t_2 := \mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+75}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-215}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, -x, x\right)\\ \mathbf{elif}\;t \leq 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z t) a) x)) (t_2 (fma a (/ (- y x) t) y)))
   (if (<= t -4.5e+75)
     t_2
     (if (<= t 6.5e-303)
       t_1
       (if (<= t 1.15e-215)
         (fma (/ z a) (- x) x)
         (if (<= t 1e+118) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((z - t) / a), x);
	double t_2 = fma(a, ((y - x) / t), y);
	double tmp;
	if (t <= -4.5e+75) {
		tmp = t_2;
	} else if (t <= 6.5e-303) {
		tmp = t_1;
	} else if (t <= 1.15e-215) {
		tmp = fma((z / a), -x, x);
	} else if (t <= 1e+118) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(z - t) / a), x)
	t_2 = fma(a, Float64(Float64(y - x) / t), y)
	tmp = 0.0
	if (t <= -4.5e+75)
		tmp = t_2;
	elseif (t <= 6.5e-303)
		tmp = t_1;
	elseif (t <= 1.15e-215)
		tmp = fma(Float64(z / a), Float64(-x), x);
	elseif (t <= 1e+118)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -4.5e+75], t$95$2, If[LessEqual[t, 6.5e-303], t$95$1, If[LessEqual[t, 1.15e-215], N[(N[(z / a), $MachinePrecision] * (-x) + x), $MachinePrecision], If[LessEqual[t, 1e+118], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.5000000000000004e75 or 9.99999999999999967e117 < t

   1. Initial program 28.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 88.5

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

   7. Applied rewrites 88.5%

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

   8. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if -4.5000000000000004e75 < t < 6.50000000000000028e-303 or 1.15e-215 < t < 9.99999999999999967e117

   1. Initial program 83.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 62.6

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

   8. Applied rewrites 62.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 68.3

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

   10. Applied rewrites 68.3%

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

   if 6.50000000000000028e-303 < t < 1.15e-215

   1. Initial program 88.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. mul-1-neg N/A

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

      12. *-lft-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 73.2

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

   8. Applied rewrites 73.2%

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

Alternative 3: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+89}:\\ \;\;\;\;y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{z - a}{\frac{t}{x - y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3e+89)
   (+ y (* (/ (- y x) t) (- a z)))
   (if (<= t 4.1e+118)
     (fma (/ (- z t) (- a t)) (- y x) x)
     (+ y (/ (- z a) (/ t (- x y)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3e+89) {
		tmp = y + (((y - x) / t) * (a - z));
	} else if (t <= 4.1e+118) {
		tmp = fma(((z - t) / (a - t)), (y - x), x);
	} else {
		tmp = y + ((z - a) / (t / (x - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3e+89)
		tmp = Float64(y + Float64(Float64(Float64(y - x) / t) * Float64(a - z)));
	elseif (t <= 4.1e+118)
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x);
	else
		tmp = Float64(y + Float64(Float64(z - a) / Float64(t / Float64(x - y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3e+89], N[(y + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.1e+118], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(y + N[(N[(z - a), $MachinePrecision] / N[(t / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.00000000000000013e89

   1. Initial program 24.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 85.3

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

   7. Applied rewrites 85.3%

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

   if -3.00000000000000013e89 < t < 4.0999999999999997e118

   1. Initial program 84.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 93.4

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

   4. Applied rewrites 93.4%

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

   if 4.0999999999999997e118 < t

   1. Initial program 29.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 90.8

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

   7. Applied rewrites 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 90.9

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

   9. Applied rewrites 90.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+89}:\\ \;\;\;\;y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{z - a}{\frac{t}{x - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-215}:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -5.8e+72)
     t_1
     (if (<= t 1.5e-215)
       (+ x (* z (/ (- y x) a)))
       (if (<= t 1.15e+46) (fma y (/ (- z t) 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 <= -5.8e+72) {
		tmp = t_1;
	} else if (t <= 1.5e-215) {
		tmp = x + (z * ((y - x) / a));
	} else if (t <= 1.15e+46) {
		tmp = fma(y, ((z - t) / 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 <= -5.8e+72)
		tmp = t_1;
	elseif (t <= 1.5e-215)
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / a)));
	elseif (t <= 1.15e+46)
		tmp = fma(y, Float64(Float64(z - t) / 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, -5.8e+72], t$95$1, If[LessEqual[t, 1.5e-215], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e+46], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.80000000000000034e72 or 1.15e46 < t

   1. Initial program 33.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. Applied rewrites 80.2%

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

   if -5.80000000000000034e72 < t < 1.50000000000000013e-215

   1. Initial program 85.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 93.1

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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.1

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

   7. Applied rewrites 76.1%

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

   if 1.50000000000000013e-215 < t < 1.15e46

   1. Initial program 87.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 69.5

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

   8. Applied rewrites 69.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 78.2

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

   10. Applied rewrites 78.2%

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

Alternative 5: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-215}:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ z t) y)))
   (if (<= t -5.8e+72)
     t_1
     (if (<= t 1.5e-215)
       (+ x (* z (/ (- y x) a)))
       (if (<= t 1.15e+46) (fma y (/ (- z t) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), (z / t), y);
	double tmp;
	if (t <= -5.8e+72) {
		tmp = t_1;
	} else if (t <= 1.5e-215) {
		tmp = x + (z * ((y - x) / a));
	} else if (t <= 1.15e+46) {
		tmp = fma(y, ((z - t) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(z / t), y)
	tmp = 0.0
	if (t <= -5.8e+72)
		tmp = t_1;
	elseif (t <= 1.5e-215)
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / a)));
	elseif (t <= 1.15e+46)
		tmp = fma(y, Float64(Float64(z - t) / 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[(z / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -5.8e+72], t$95$1, If[LessEqual[t, 1.5e-215], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e+46], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.80000000000000034e72 or 1.15e46 < t

   1. Initial program 33.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 84.8

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 72.3

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

   10. Applied rewrites 72.3%

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

   if -5.80000000000000034e72 < t < 1.50000000000000013e-215

   1. Initial program 85.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 93.1

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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.1

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

   7. Applied rewrites 76.1%

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

   if 1.50000000000000013e-215 < t < 1.15e46

   1. Initial program 87.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 69.5

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

   8. Applied rewrites 69.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 78.2

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

   10. Applied rewrites 78.2%

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

Alternative 6: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-215}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ z t) y)))
   (if (<= t -5.8e+72)
     t_1
     (if (<= t 1.5e-215)
       (fma z (/ (- y x) a) x)
       (if (<= t 1.15e+46) (fma y (/ (- z t) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), (z / t), y);
	double tmp;
	if (t <= -5.8e+72) {
		tmp = t_1;
	} else if (t <= 1.5e-215) {
		tmp = fma(z, ((y - x) / a), x);
	} else if (t <= 1.15e+46) {
		tmp = fma(y, ((z - t) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(z / t), y)
	tmp = 0.0
	if (t <= -5.8e+72)
		tmp = t_1;
	elseif (t <= 1.5e-215)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	elseif (t <= 1.15e+46)
		tmp = fma(y, Float64(Float64(z - t) / 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[(z / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -5.8e+72], t$95$1, If[LessEqual[t, 1.5e-215], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.15e+46], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.80000000000000034e72 or 1.15e46 < t

   1. Initial program 33.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 84.8

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 72.3

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

   10. Applied rewrites 72.3%

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

   if -5.80000000000000034e72 < t < 1.50000000000000013e-215

   1. Initial program 85.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 76.1

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

   5. Applied rewrites 76.1%

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

   if 1.50000000000000013e-215 < t < 1.15e46

   1. Initial program 87.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 69.5

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

   8. Applied rewrites 69.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 78.2

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

   10. Applied rewrites 78.2%

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

Alternative 7: 62.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-215}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;t \leq 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- y x) t) y)))
   (if (<= t -4.5e+75)
     t_1
     (if (<= t 1.5e-215)
       (fma z (/ (- y x) a) x)
       (if (<= t 1e+118) (fma y (/ (- z t) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((y - x) / t), y);
	double tmp;
	if (t <= -4.5e+75) {
		tmp = t_1;
	} else if (t <= 1.5e-215) {
		tmp = fma(z, ((y - x) / a), x);
	} else if (t <= 1e+118) {
		tmp = fma(y, ((z - t) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(y - x) / t), y)
	tmp = 0.0
	if (t <= -4.5e+75)
		tmp = t_1;
	elseif (t <= 1.5e-215)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	elseif (t <= 1e+118)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -4.5e+75], t$95$1, If[LessEqual[t, 1.5e-215], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1e+118], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.5000000000000004e75 or 9.99999999999999967e117 < t

   1. Initial program 28.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 88.5

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

   7. Applied rewrites 88.5%

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

   8. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if -4.5000000000000004e75 < t < 1.50000000000000013e-215

   1. Initial program 85.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 76.1

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

   5. Applied rewrites 76.1%

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

   if 1.50000000000000013e-215 < t < 9.99999999999999967e117

   1. Initial program 83.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 62.7

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

   8. Applied rewrites 62.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

Alternative 8: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \mathbf{if}\;t \leq -3 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ y (* (/ (- y x) t) (- a z)))))
   (if (<= t -3e+89)
     t_1
     (if (<= t 4.1e+118) (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 = y + (((y - x) / t) * (a - z));
	double tmp;
	if (t <= -3e+89) {
		tmp = t_1;
	} else if (t <= 4.1e+118) {
		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 = Float64(y + Float64(Float64(Float64(y - x) / t) * Float64(a - z)))
	tmp = 0.0
	if (t <= -3e+89)
		tmp = t_1;
	elseif (t <= 4.1e+118)
		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[(y + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3e+89], t$95$1, If[LessEqual[t, 4.1e+118], 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 < -3.00000000000000013e89 or 4.0999999999999997e118 < t

   1. Initial program 26.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 28.8%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 88.2

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

   7. Applied rewrites 88.2%

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

   if -3.00000000000000013e89 < t < 4.0999999999999997e118

   1. Initial program 84.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 93.4

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

   4. Applied rewrites 93.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+89}:\\ \;\;\;\;y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y - x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ y (* (/ (- y x) t) (- a z)))))
   (if (<= t -2.1e+73)
     t_1
     (if (<= t 1.15e+46) (+ x (* (- z t) (/ (- y x) a))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y + (((y - x) / t) * (a - z));
	double tmp;
	if (t <= -2.1e+73) {
		tmp = t_1;
	} else if (t <= 1.15e+46) {
		tmp = x + ((z - t) * ((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 + (((y - x) / t) * (a - z))
    if (t <= (-2.1d+73)) then
        tmp = t_1
    else if (t <= 1.15d+46) then
        tmp = x + ((z - t) * ((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 + (((y - x) / t) * (a - z));
	double tmp;
	if (t <= -2.1e+73) {
		tmp = t_1;
	} else if (t <= 1.15e+46) {
		tmp = x + ((z - t) * ((y - x) / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y + (((y - x) / t) * (a - z))
	tmp = 0
	if t <= -2.1e+73:
		tmp = t_1
	elif t <= 1.15e+46:
		tmp = x + ((z - t) * ((y - x) / a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y + Float64(Float64(Float64(y - x) / t) * Float64(a - z)))
	tmp = 0.0
	if (t <= -2.1e+73)
		tmp = t_1;
	elseif (t <= 1.15e+46)
		tmp = Float64(x + Float64(Float64(z - t) * 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 + (((y - x) / t) * (a - z));
	tmp = 0.0;
	if (t <= -2.1e+73)
		tmp = t_1;
	elseif (t <= 1.15e+46)
		tmp = x + ((z - t) * ((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[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.1e+73], t$95$1, If[LessEqual[t, 1.15e+46], N[(x + N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1000000000000001e73 or 1.15e46 < t

   1. Initial program 33.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 84.8

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

   7. Applied rewrites 84.8%

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

   if -2.1000000000000001e73 < t < 1.15e46

   1. Initial program 85.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

      \[\leadsto x + \color{blue}{\left(z - t\right) \cdot \frac{y - x}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+73}:\\ \;\;\;\;y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y - x}{a}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{y - x}{t} \cdot \left(a - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y - x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -2.1e+73)
     t_1
     (if (<= t 1.15e+46) (+ x (* (- z t) (/ (- y x) a))) 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 <= -2.1e+73) {
		tmp = t_1;
	} else if (t <= 1.15e+46) {
		tmp = x + ((z - t) * ((y - x) / a));
	} 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 <= -2.1e+73)
		tmp = t_1;
	elseif (t <= 1.15e+46)
		tmp = Float64(x + Float64(Float64(z - t) * Float64(Float64(y - x) / a)));
	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, -2.1e+73], t$95$1, If[LessEqual[t, 1.15e+46], N[(x + N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1000000000000001e73 or 1.15e46 < t

   1. Initial program 33.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. Applied rewrites 80.2%

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

   if -2.1000000000000001e73 < t < 1.15e46

   1. Initial program 85.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

Alternative 11: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -2.1e+73)
     t_1
     (if (<= t 1.15e+46) (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 <= -2.1e+73) {
		tmp = t_1;
	} else if (t <= 1.15e+46) {
		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 <= -2.1e+73)
		tmp = t_1;
	elseif (t <= 1.15e+46)
		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, -2.1e+73], t$95$1, If[LessEqual[t, 1.15e+46], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1000000000000001e73 or 1.15e46 < t

   1. Initial program 33.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. Applied rewrites 80.2%

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

   if -2.1000000000000001e73 < t < 1.15e46

   1. Initial program 85.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

Alternative 12: 55.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{if}\;t \leq -4 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- y x) t) y)))
   (if (<= t -4e+75) t_1 (if (<= t 9.6e+117) (fma z (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((y - x) / t), y);
	double tmp;
	if (t <= -4e+75) {
		tmp = t_1;
	} else if (t <= 9.6e+117) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(y - x) / t), y)
	tmp = 0.0
	if (t <= -4e+75)
		tmp = t_1;
	elseif (t <= 9.6e+117)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -4e+75], t$95$1, If[LessEqual[t, 9.6e+117], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.99999999999999971e75 or 9.5999999999999996e117 < t

   1. Initial program 28.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 88.5

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

   7. Applied rewrites 88.5%

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

   8. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if -3.99999999999999971e75 < t < 9.5999999999999996e117

   1. Initial program 84.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 57.4

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

   8. Applied rewrites 57.4%

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

Alternative 13: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x - x\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ y (- x x))))
   (if (<= t -2.8e+143) t_1 (if (<= t 9.6e+117) (fma z (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y + (x - x);
	double tmp;
	if (t <= -2.8e+143) {
		tmp = t_1;
	} else if (t <= 9.6e+117) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y + Float64(x - x))
	tmp = 0.0
	if (t <= -2.8e+143)
		tmp = t_1;
	elseif (t <= 9.6e+117)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y + N[(x - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e+143], t$95$1, If[LessEqual[t, 9.6e+117], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.79999999999999998e143 or 9.5999999999999996e117 < t

   1. Initial program 23.3%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 35.8

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

   5. Applied rewrites 35.8%

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   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 56.5

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

   7. Applied rewrites 56.5%

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

   if -2.79999999999999998e143 < t < 9.5999999999999996e117

   1. Initial program 82.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 55.8

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

   8. Applied rewrites 55.8%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+143}:\\ \;\;\;\;y + \left(x - x\right)\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x - x\right)\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ y (- x x))))
   (if (<= t -2.1e+79) t_1 (if (<= t 1.45e+48) (* y (/ z a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y + (x - x);
	double tmp;
	if (t <= -2.1e+79) {
		tmp = t_1;
	} else if (t <= 1.45e+48) {
		tmp = y * (z / 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 + (x - x)
    if (t <= (-2.1d+79)) then
        tmp = t_1
    else if (t <= 1.45d+48) then
        tmp = y * (z / 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 + (x - x);
	double tmp;
	if (t <= -2.1e+79) {
		tmp = t_1;
	} else if (t <= 1.45e+48) {
		tmp = y * (z / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y + (x - x)
	tmp = 0
	if t <= -2.1e+79:
		tmp = t_1
	elif t <= 1.45e+48:
		tmp = y * (z / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y + Float64(x - x))
	tmp = 0.0
	if (t <= -2.1e+79)
		tmp = t_1;
	elseif (t <= 1.45e+48)
		tmp = Float64(y * Float64(z / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y + (x - x);
	tmp = 0.0;
	if (t <= -2.1e+79)
		tmp = t_1;
	elseif (t <= 1.45e+48)
		tmp = y * (z / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y + N[(x - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.1e+79], t$95$1, If[LessEqual[t, 1.45e+48], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.10000000000000008e79 or 1.4499999999999999e48 < t

   1. Initial program 31.1%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 32.4

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

   5. Applied rewrites 32.4%

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   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 50.0

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

   7. Applied rewrites 50.0%

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

   if -2.10000000000000008e79 < t < 1.4499999999999999e48

   1. Initial program 86.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 28.7

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

   8. Applied rewrites 28.7%

      \[\leadsto \color{blue}{y \cdot \frac{z}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+79}:\\ \;\;\;\;y + \left(x - x\right)\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ y + \left(x - x\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
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 17.4

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

5. Applied rewrites 17.4%

   \[\leadsto x + \color{blue}{\left(y - x\right)} \]
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 24.1

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

7. Applied rewrites 24.1%

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

8. Final simplification 24.1%

   \[\leadsto y + \left(x - x\right) \]
9. Add Preprocessing

Alternative 16: 19.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ x + \left(y - x\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
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 17.4

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

5. Applied rewrites 17.4%

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

Alternative 17: 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 66.2%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
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 17.4

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

5. Applied rewrites 17.4%

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

6. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 2.8

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

8. Applied rewrites 2.8%

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

   1. unsub-neg N/A

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

   2. +-inverses 2.8

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

10. Applied rewrites 2.8%

    \[\leadsto \color{blue}{0} \]
11. 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 2024219 
(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))))