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

Percentage Accurate: 68.2% → 89.2%

Time: 13.7s

Alternatives: 18

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.2% 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: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{t}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\left(y + a \cdot \frac{y - x}{t}\right) + z \cdot t\_1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+118}:\\ \;\;\;\;x + \frac{x - y}{\frac{a - t}{t - z}}\\ \mathbf{else}:\\ \;\;\;\;y + t\_1 \cdot \left(z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x y) t)))
   (if (<= t -2e+145)
     (+ (+ y (* a (/ (- y x) t))) (* z t_1))
     (if (<= t 4.1e+118)
       (+ x (/ (- x y) (/ (- a t) (- t z))))
       (+ y (* t_1 (- z a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - y) / t;
	double tmp;
	if (t <= -2e+145) {
		tmp = (y + (a * ((y - x) / t))) + (z * t_1);
	} else if (t <= 4.1e+118) {
		tmp = x + ((x - y) / ((a - t) / (t - z)));
	} else {
		tmp = y + (t_1 * (z - a));
	}
	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) / t
    if (t <= (-2d+145)) then
        tmp = (y + (a * ((y - x) / t))) + (z * t_1)
    else if (t <= 4.1d+118) then
        tmp = x + ((x - y) / ((a - t) / (t - z)))
    else
        tmp = y + (t_1 * (z - a))
    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) / t;
	double tmp;
	if (t <= -2e+145) {
		tmp = (y + (a * ((y - x) / t))) + (z * t_1);
	} else if (t <= 4.1e+118) {
		tmp = x + ((x - y) / ((a - t) / (t - z)));
	} else {
		tmp = y + (t_1 * (z - a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - y) / t
	tmp = 0
	if t <= -2e+145:
		tmp = (y + (a * ((y - x) / t))) + (z * t_1)
	elif t <= 4.1e+118:
		tmp = x + ((x - y) / ((a - t) / (t - z)))
	else:
		tmp = y + (t_1 * (z - a))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - y) / t)
	tmp = 0.0
	if (t <= -2e+145)
		tmp = Float64(Float64(y + Float64(a * Float64(Float64(y - x) / t))) + Float64(z * t_1));
	elseif (t <= 4.1e+118)
		tmp = Float64(x + Float64(Float64(x - y) / Float64(Float64(a - t) / Float64(t - z))));
	else
		tmp = Float64(y + Float64(t_1 * Float64(z - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - y) / t;
	tmp = 0.0;
	if (t <= -2e+145)
		tmp = (y + (a * ((y - x) / t))) + (z * t_1);
	elseif (t <= 4.1e+118)
		tmp = x + ((x - y) / ((a - t) / (t - z)));
	else
		tmp = y + (t_1 * (z - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -2e+145], N[(N[(y + N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.1e+118], N[(x + N[(N[(x - y), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(t$95$1 * N[(z - a), $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. 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 89.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 93.0%

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

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. 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

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 12.9

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

   4. Applied rewrites 12.9%

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

   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%

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

4. Final simplification 91.9%

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

Alternative 2: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \frac{x - y}{t} \cdot \left(z - a\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+118}:\\ \;\;\;\;x + \frac{x - y}{\frac{a - t}{t - z}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ y (* (/ (- x y) t) (- z a)))))
   (if (<= t -2e+145)
     t_1
     (if (<= t 4.1e+118) (+ x (/ (- x y) (/ (- a t) (- t z)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y + (((x - y) / t) * (z - a));
	double tmp;
	if (t <= -2e+145) {
		tmp = t_1;
	} else if (t <= 4.1e+118) {
		tmp = x + ((x - y) / ((a - t) / (t - z)));
	} 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 - y) / t) * (z - a))
    if (t <= (-2d+145)) then
        tmp = t_1
    else if (t <= 4.1d+118) then
        tmp = x + ((x - y) / ((a - t) / (t - z)))
    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 - y) / t) * (z - a));
	double tmp;
	if (t <= -2e+145) {
		tmp = t_1;
	} else if (t <= 4.1e+118) {
		tmp = x + ((x - y) / ((a - t) / (t - z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y + (((x - y) / t) * (z - a))
	tmp = 0
	if t <= -2e+145:
		tmp = t_1
	elif t <= 4.1e+118:
		tmp = x + ((x - y) / ((a - t) / (t - z)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y + Float64(Float64(Float64(x - y) / t) * Float64(z - a)))
	tmp = 0.0
	if (t <= -2e+145)
		tmp = t_1;
	elseif (t <= 4.1e+118)
		tmp = Float64(x + Float64(Float64(x - y) / Float64(Float64(a - t) / Float64(t - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y + (((x - y) / t) * (z - a));
	tmp = 0.0;
	if (t <= -2e+145)
		tmp = t_1;
	elseif (t <= 4.1e+118)
		tmp = x + ((x - y) / ((a - t) / (t - z)));
	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[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+145], t$95$1, If[LessEqual[t, 4.1e+118], N[(x + N[(N[(x - y), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2e145 or 4.0999999999999997e118 < t

   1. Initial program 23.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 8.9

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

   4. Applied rewrites 8.9%

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

   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 91.7

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

   7. Applied rewrites 91.7%

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

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.0

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

   4. Applied rewrites 92.0%

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

4. Final simplification 91.9%

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

Alternative 3: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -5.8e+72)
     t_1
     (if (<= t 2.4e-161)
       (fma z (/ (- y x) a) x)
       (if (<= t 1.15e+46) (+ x (* (- z t) (/ y 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 <= -5.8e+72) {
		tmp = t_1;
	} else if (t <= 2.4e-161) {
		tmp = fma(z, ((y - x) / a), x);
	} else if (t <= 1.15e+46) {
		tmp = x + ((z - t) * (y / 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 <= -5.8e+72)
		tmp = t_1;
	elseif (t <= 2.4e-161)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	elseif (t <= 1.15e+46)
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / 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, -5.8e+72], t$95$1, If[LessEqual[t, 2.4e-161], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.15e+46], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $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 < 2.39999999999999999e-161

   1. Initial program 85.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 75.7

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

   5. Applied rewrites 75.7%

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

   if 2.39999999999999999e-161 < t < 1.15e46

   1. Initial program 86.3%

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

   3. Taylor expanded in 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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 75.4%

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

Alternative 4: 68.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ z t) y)))
   (if (<= t -5.8e+72)
     t_1
     (if (<= t 2.4e-161)
       (fma z (/ (- y x) a) x)
       (if (<= t 1.15e+46) (+ x (* (- z t) (/ y a))) 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 <= 2.4e-161) {
		tmp = fma(z, ((y - x) / a), x);
	} else if (t <= 1.15e+46) {
		tmp = x + ((z - t) * (y / a));
	} 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 <= 2.4e-161)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	elseif (t <= 1.15e+46)
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / 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[(z / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -5.8e+72], t$95$1, If[LessEqual[t, 2.4e-161], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.15e+46], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $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%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 72.3%

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

   if -5.80000000000000034e72 < t < 2.39999999999999999e-161

   1. Initial program 85.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 75.7

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

   5. Applied rewrites 75.7%

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

   if 2.39999999999999999e-161 < t < 1.15e46

   1. Initial program 86.3%

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

   3. Taylor expanded in 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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 75.4%

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

Alternative 5: 88.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ y (* (/ (- x y) t) (- z a)))))
   (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 + (((x - y) / t) * (z - a));
	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(x - y) / t) * Float64(z - a)))
	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[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $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

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 14.1

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

   4. Applied rewrites 14.1%

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

   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

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. 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%

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

Alternative 6: 75.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 21.5

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

   4. Applied rewrites 21.5%

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

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 79.1

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

   7. Applied rewrites 79.1%

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

4. Final simplification 81.2%

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

Alternative 7: 75.8% accurate, 0.8× speedup

Math

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

FPCore

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 79.1

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

   7. Applied rewrites 79.1%

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

Alternative 8: 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 9: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ z t) y)))
   (if (<= t -5.8e+72) t_1 (if (<= t 1.1e+46) (fma z (/ (- 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 / t), y);
	double tmp;
	if (t <= -5.8e+72) {
		tmp = t_1;
	} else if (t <= 1.1e+46) {
		tmp = fma(z, ((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(z / t), y)
	tmp = 0.0
	if (t <= -5.8e+72)
		tmp = t_1;
	elseif (t <= 1.1e+46)
		tmp = fma(z, 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[(z / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -5.8e+72], t$95$1, If[LessEqual[t, 1.1e+46], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.80000000000000034e72 or 1.1e46 < 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%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 72.3%

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

   if -5.80000000000000034e72 < t < 1.1e46

   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 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 72.6

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

   5. Applied rewrites 72.6%

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

Alternative 10: 62.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y) (/ z t) y)))
   (if (<= t -1.6e+75) t_1 (if (<= t 9.6e+117) (fma z (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-y, (z / t), y);
	double tmp;
	if (t <= -1.6e+75) {
		tmp = t_1;
	} else if (t <= 9.6e+117) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-y), Float64(z / t), y)
	tmp = 0.0
	if (t <= -1.6e+75)
		tmp = t_1;
	elseif (t <= 9.6e+117)
		tmp = fma(z, 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[((-y) * N[(z / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.6e+75], t$95$1, If[LessEqual[t, 9.6e+117], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.59999999999999992e75 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. 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 83.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 61.8%

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

   if -1.59999999999999992e75 < 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%

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

Alternative 11: 53.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+46}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y) (/ z t) y)))
   (if (<= t -8.5e+74) t_1 (if (<= t 2.2e+46) (+ x (/ (* y z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-y, (z / t), y);
	double tmp;
	if (t <= -8.5e+74) {
		tmp = t_1;
	} else if (t <= 2.2e+46) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-y), Float64(z / t), y)
	tmp = 0.0
	if (t <= -8.5e+74)
		tmp = t_1;
	elseif (t <= 2.2e+46)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-y) * N[(z / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -8.5e+74], t$95$1, If[LessEqual[t, 2.2e+46], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.50000000000000028e74 or 2.2e46 < 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%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 72.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.4%

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

   if -8.50000000000000028e74 < t < 2.2e46

   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 t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 55.5%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z}{t}, y\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+46}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y) (/ z t) y)))
   (if (<= t -5.8e+72) t_1 (if (<= t 2e+46) (* y (/ (- z t) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-y, (z / t), y);
	double tmp;
	if (t <= -5.8e+72) {
		tmp = t_1;
	} else if (t <= 2e+46) {
		tmp = y * ((z - t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-y), Float64(z / t), y)
	tmp = 0.0
	if (t <= -5.8e+72)
		tmp = t_1;
	elseif (t <= 2e+46)
		tmp = Float64(y * Float64(Float64(z - t) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-y) * N[(z / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -5.8e+72], t$95$1, If[LessEqual[t, 2e+46], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

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

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 72.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.4%

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

   if -5.80000000000000034e72 < t < 2e46

   1. Initial program 85.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 40.6

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

   7. Applied rewrites 40.6%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 34.6%

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

Alternative 13: 39.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(z - a\right)}{t}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* x (- z a)) t)))
   (if (<= x -7.8e+182) t_1 (if (<= x 7e+43) (fma (- y) (/ z t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * (z - a)) / t;
	double tmp;
	if (x <= -7.8e+182) {
		tmp = t_1;
	} else if (x <= 7e+43) {
		tmp = fma(-y, (z / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * Float64(z - a)) / t)
	tmp = 0.0
	if (x <= -7.8e+182)
		tmp = t_1;
	elseif (x <= 7e+43)
		tmp = fma(Float64(-y), Float64(z / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[x, -7.8e+182], t$95$1, If[LessEqual[x, 7e+43], N[((-y) * N[(z / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.7999999999999998e182 or 7.0000000000000002e43 < x

   1. Initial program 55.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 48.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 38.6%

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

   if -7.7999999999999998e182 < x < 7.0000000000000002e43

   1. Initial program 71.5%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 43.4%

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

Alternative 14: 31.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{x - y}{t}\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-118}:\\ \;\;\;\;x + \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ (- x y) t))))
   (if (<= z -1.32e+101) t_1 (if (<= z 5.5e-118) (+ x (- y x)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * ((x - y) / t);
	double tmp;
	if (z <= -1.32e+101) {
		tmp = t_1;
	} else if (z <= 5.5e-118) {
		tmp = x + (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 = z * ((x - y) / t)
    if (z <= (-1.32d+101)) then
        tmp = t_1
    else if (z <= 5.5d-118) then
        tmp = x + (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 = z * ((x - y) / t);
	double tmp;
	if (z <= -1.32e+101) {
		tmp = t_1;
	} else if (z <= 5.5e-118) {
		tmp = x + (y - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * ((x - y) / t)
	tmp = 0
	if z <= -1.32e+101:
		tmp = t_1
	elif z <= 5.5e-118:
		tmp = x + (y - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(Float64(x - y) / t))
	tmp = 0.0
	if (z <= -1.32e+101)
		tmp = t_1;
	elseif (z <= 5.5e-118)
		tmp = Float64(x + Float64(y - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * ((x - y) / t);
	tmp = 0.0;
	if (z <= -1.32e+101)
		tmp = t_1;
	elseif (z <= 5.5e-118)
		tmp = x + (y - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.32e+101], t$95$1, If[LessEqual[z, 5.5e-118], N[(x + N[(y - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e101 or 5.5000000000000003e-118 < z

   1. Initial program 67.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 41.7%

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

   if -1.32e101 < z < 5.5000000000000003e-118

   1. Initial program 65.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 27.0

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

   5. Applied rewrites 27.0%

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

Alternative 15: 26.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+68}:\\ \;\;\;\;x + \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e+93)
   (* y (/ z a))
   (if (<= z 6.5e+68) (+ x (- y x)) (* z (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e+93) {
		tmp = y * (z / a);
	} else if (z <= 6.5e+68) {
		tmp = x + (y - x);
	} else {
		tmp = z * (y / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.3d+93)) then
        tmp = y * (z / a)
    else if (z <= 6.5d+68) then
        tmp = x + (y - x)
    else
        tmp = z * (y / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e+93) {
		tmp = y * (z / a);
	} else if (z <= 6.5e+68) {
		tmp = x + (y - x);
	} else {
		tmp = z * (y / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.3e+93:
		tmp = y * (z / a)
	elif z <= 6.5e+68:
		tmp = x + (y - x)
	else:
		tmp = z * (y / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.3e+93)
		tmp = Float64(y * Float64(z / a));
	elseif (z <= 6.5e+68)
		tmp = Float64(x + Float64(y - x));
	else
		tmp = Float64(z * Float64(y / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.3e+93)
		tmp = y * (z / a);
	elseif (z <= 6.5e+68)
		tmp = x + (y - x);
	else
		tmp = z * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.3e+93], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e+68], N[(x + N[(y - x), $MachinePrecision]), $MachinePrecision], N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e93

   1. Initial program 63.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 49.5

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 35.3

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

   7. Applied rewrites 35.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 29.2%

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

   12. Applied rewrites 40.3%

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

   if -1.3e93 < z < 6.5000000000000005e68

   1. Initial program 64.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 24.9

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

   5. Applied rewrites 24.9%

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

   if 6.5000000000000005e68 < z

   1. Initial program 75.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 48.4

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 32.2%

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

   12. Applied rewrites 37.9%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+68}:\\ \;\;\;\;x + \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{a}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+68}:\\ \;\;\;\;x + \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ y a))))
   (if (<= z -1.05e+93) t_1 (if (<= z 6.5e+68) (+ x (- y x)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (y / a);
	double tmp;
	if (z <= -1.05e+93) {
		tmp = t_1;
	} else if (z <= 6.5e+68) {
		tmp = x + (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 = z * (y / a)
    if (z <= (-1.05d+93)) then
        tmp = t_1
    else if (z <= 6.5d+68) then
        tmp = x + (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 = z * (y / a);
	double tmp;
	if (z <= -1.05e+93) {
		tmp = t_1;
	} else if (z <= 6.5e+68) {
		tmp = x + (y - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * (y / a)
	tmp = 0
	if z <= -1.05e+93:
		tmp = t_1
	elif z <= 6.5e+68:
		tmp = x + (y - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(y / a))
	tmp = 0.0
	if (z <= -1.05e+93)
		tmp = t_1;
	elseif (z <= 6.5e+68)
		tmp = Float64(x + Float64(y - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * (y / a);
	tmp = 0.0;
	if (z <= -1.05e+93)
		tmp = t_1;
	elseif (z <= 6.5e+68)
		tmp = x + (y - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e+93], t$95$1, If[LessEqual[z, 6.5e+68], N[(x + N[(y - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0499999999999999e93 or 6.5000000000000005e68 < z

   1. Initial program 69.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 41.7

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 30.6%

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

   12. Applied rewrites 38.4%

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

   if -1.0499999999999999e93 < z < 6.5000000000000005e68

   1. Initial program 64.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 24.9

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

   5. Applied rewrites 24.9%

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

Alternative 17: 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 18: 2.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + (-x)), $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. Taylor expanded in y around 0

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

8. Applied rewrites 2.8%

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

Developer Target 1: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 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))))