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

Percentage Accurate: 68.4% → 89.3%

Time: 12.5s

Alternatives: 16

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: 68.4% 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: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.32 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+152}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.32e+109)
     t_1
     (if (<= t 7.5e+152) (+ x (/ (- y x) (/ (- a t) (- z t)))) 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 <= -1.32e+109) {
		tmp = t_1;
	} else if (t <= 7.5e+152) {
		tmp = x + ((y - x) / ((a - t) / (z - t)));
	} 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 <= -1.32e+109)
		tmp = t_1;
	elseif (t <= 7.5e+152)
		tmp = Float64(x + Float64(Float64(y - x) / Float64(Float64(a - t) / Float64(z - t))));
	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, -1.32e+109], t$95$1, If[LessEqual[t, 7.5e+152], N[(x + N[(N[(y - x), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.32000000000000008e109 or 7.50000000000000046e152 < t

   1. Initial program 30.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   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 90.8%

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

   if -1.32000000000000008e109 < t < 7.50000000000000046e152

   1. Initial program 83.4%

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

      1. lift-/.f64 N/A

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

      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. Add Preprocessing

Alternative 2: 47.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{x - y}{a}, x\right)\\ t_2 := \mathsf{fma}\left(\frac{z}{t}, -y, y\right)\\ \mathbf{if}\;t \leq -250:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-117}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ (- x y) a) x)) (t_2 (fma (/ z t) (- y) y)))
   (if (<= t -250.0)
     t_2
     (if (<= t -5.2e-116)
       t_1
       (if (<= t 1.02e-117)
         (/ (* z (- y x)) a)
         (if (<= t 1.9e-43)
           t_1
           (if (<= t 9.5e+41) (/ (* z (- x y)) t) t_2)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, ((x - y) / a), x);
	double t_2 = fma((z / t), -y, y);
	double tmp;
	if (t <= -250.0) {
		tmp = t_2;
	} else if (t <= -5.2e-116) {
		tmp = t_1;
	} else if (t <= 1.02e-117) {
		tmp = (z * (y - x)) / a;
	} else if (t <= 1.9e-43) {
		tmp = t_1;
	} else if (t <= 9.5e+41) {
		tmp = (z * (x - y)) / t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(Float64(x - y) / a), x)
	t_2 = fma(Float64(z / t), Float64(-y), y)
	tmp = 0.0
	if (t <= -250.0)
		tmp = t_2;
	elseif (t <= -5.2e-116)
		tmp = t_1;
	elseif (t <= 1.02e-117)
		tmp = Float64(Float64(z * Float64(y - x)) / a);
	elseif (t <= 1.9e-43)
		tmp = t_1;
	elseif (t <= 9.5e+41)
		tmp = Float64(Float64(z * Float64(x - y)) / t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(x - y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / t), $MachinePrecision] * (-y) + y), $MachinePrecision]}, If[LessEqual[t, -250.0], t$95$2, If[LessEqual[t, -5.2e-116], t$95$1, If[LessEqual[t, 1.02e-117], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 1.9e-43], t$95$1, If[LessEqual[t, 9.5e+41], N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -250 or 9.4999999999999996e41 < t

   1. Initial program 44.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 12.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.2%

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

   if -250 < t < -5.2000000000000001e-116 or 1.01999999999999993e-117 < t < 1.89999999999999985e-43

   1. Initial program 89.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 55.9%

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

   if -5.2000000000000001e-116 < t < 1.01999999999999993e-117

   1. Initial program 93.9%

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

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 64.0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 54.8%

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

   if 1.89999999999999985e-43 < t < 9.4999999999999996e41

   1. Initial program 80.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 61.6

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

   5. Applied rewrites 61.6%

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

   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 55.9%

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

Alternative 3: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.32 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{1}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.32e+109)
     t_1
     (if (<= t 7.5e+152) (fma (* (- z t) (/ 1.0 (- a t))) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -1.32e+109) {
		tmp = t_1;
	} else if (t <= 7.5e+152) {
		tmp = fma(((z - t) * (1.0 / (a - t))), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -1.32e+109)
		tmp = t_1;
	elseif (t <= 7.5e+152)
		tmp = fma(Float64(Float64(z - t) * Float64(1.0 / Float64(a - t))), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.32e+109], t$95$1, If[LessEqual[t, 7.5e+152], N[(N[(N[(z - t), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.32000000000000008e109 or 7.50000000000000046e152 < t

   1. Initial program 30.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   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 90.8%

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

   if -1.32000000000000008e109 < t < 7.50000000000000046e152

   1. Initial program 83.4%

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

      1. lift-+.f64 N/A

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

      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. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 90.9

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

   4. Applied rewrites 90.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{1}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+265}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- x y) t) y)))
   (if (<= t -2.7e+265)
     (fma a (/ (- y x) t) y)
     (if (<= t -1.8e+53)
       t_1
       (if (<= t 4.6e-43) (fma (/ z a) (- y x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((x - y) / t), y);
	double tmp;
	if (t <= -2.7e+265) {
		tmp = fma(a, ((y - x) / t), y);
	} else if (t <= -1.8e+53) {
		tmp = t_1;
	} else if (t <= 4.6e-43) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(x - y) / t), y)
	tmp = 0.0
	if (t <= -2.7e+265)
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	elseif (t <= -1.8e+53)
		tmp = t_1;
	elseif (t <= 4.6e-43)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.7e+265], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t, -1.8e+53], t$95$1, If[LessEqual[t, 4.6e-43], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.69999999999999984e265

   1. Initial program 11.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 94.1%

      \[\leadsto \left(0 + a \cdot \frac{y}{t}\right) - \color{blue}{\left(a \cdot \frac{x}{t} - y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.1%

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

   if -2.69999999999999984e265 < t < -1.8e53 or 4.5999999999999998e-43 < t

   1. Initial program 50.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 74.1%

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

   if -1.8e53 < t < 4.5999999999999998e-43

   1. Initial program 90.1%

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

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

      5. associate-/r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 75.5

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

   7. Applied rewrites 75.5%

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

Alternative 5: 69.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+265}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- x y) t) y)))
   (if (<= t -2.7e+265)
     (fma a (/ (- y x) t) y)
     (if (<= t -1.8e+53)
       t_1
       (if (<= t 4.6e-43) (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(z, ((x - y) / t), y);
	double tmp;
	if (t <= -2.7e+265) {
		tmp = fma(a, ((y - x) / t), y);
	} else if (t <= -1.8e+53) {
		tmp = t_1;
	} else if (t <= 4.6e-43) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(x - y) / t), y)
	tmp = 0.0
	if (t <= -2.7e+265)
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	elseif (t <= -1.8e+53)
		tmp = t_1;
	elseif (t <= 4.6e-43)
		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[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.7e+265], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t, -1.8e+53], t$95$1, If[LessEqual[t, 4.6e-43], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.69999999999999984e265

   1. Initial program 11.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 94.1%

      \[\leadsto \left(0 + a \cdot \frac{y}{t}\right) - \color{blue}{\left(a \cdot \frac{x}{t} - y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.1%

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

   if -2.69999999999999984e265 < t < -1.8e53 or 4.5999999999999998e-43 < t

   1. Initial program 50.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 74.1%

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

   if -1.8e53 < t < 4.5999999999999998e-43

   1. Initial program 90.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 72.4

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

   5. Applied rewrites 72.4%

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

Alternative 6: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.32 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.32e+109)
     t_1
     (if (<= t 5.2e+71) (fma (- z t) (/ (- y x) (- a t)) 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 <= -1.32e+109) {
		tmp = t_1;
	} else if (t <= 5.2e+71) {
		tmp = fma((z - t), ((y - x) / (a - t)), 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 <= -1.32e+109)
		tmp = t_1;
	elseif (t <= 5.2e+71)
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / Float64(a - t)), 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, -1.32e+109], t$95$1, If[LessEqual[t, 5.2e+71], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.32000000000000008e109 or 5.19999999999999983e71 < t

   1. Initial program 37.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 88.6%

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

   if -1.32000000000000008e109 < t < 5.19999999999999983e71

   1. Initial program 85.6%

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 89.5

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

   4. Applied rewrites 89.5%

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

Alternative 7: 44.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, -y, y\right)\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z t) (- y) y)))
   (if (<= t -2.4e-77)
     t_1
     (if (<= t 2.5e-85)
       (/ (* z (- y x)) a)
       (if (<= t 9.5e+41) (/ (* z (- x y)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / t), -y, y);
	double tmp;
	if (t <= -2.4e-77) {
		tmp = t_1;
	} else if (t <= 2.5e-85) {
		tmp = (z * (y - x)) / a;
	} else if (t <= 9.5e+41) {
		tmp = (z * (x - y)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / t), Float64(-y), y)
	tmp = 0.0
	if (t <= -2.4e-77)
		tmp = t_1;
	elseif (t <= 2.5e-85)
		tmp = Float64(Float64(z * Float64(y - x)) / a);
	elseif (t <= 9.5e+41)
		tmp = Float64(Float64(z * Float64(x - y)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * (-y) + y), $MachinePrecision]}, If[LessEqual[t, -2.4e-77], t$95$1, If[LessEqual[t, 2.5e-85], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 9.5e+41], N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.3999999999999999e-77 or 9.4999999999999996e41 < t

   1. Initial program 46.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 12.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 57.0%

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

   if -2.3999999999999999e-77 < t < 2.5000000000000001e-85

   1. Initial program 95.3%

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

      1. lift-+.f64 N/A

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

      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. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 59.6

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

   7. Applied rewrites 59.6%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 49.8%

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

   if 2.5000000000000001e-85 < t < 9.4999999999999996e41

   1. Initial program 86.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 52.2

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

   5. Applied rewrites 52.2%

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

   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 48.1%

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

Alternative 8: 42.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, -y, y\right)\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{-89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-85}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z t) (- y) y)))
   (if (<= t -2.3e-89)
     t_1
     (if (<= t 1.45e-85)
       (* y (/ z (- a t)))
       (if (<= t 9.5e+41) (/ (* z (- x y)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / t), -y, y);
	double tmp;
	if (t <= -2.3e-89) {
		tmp = t_1;
	} else if (t <= 1.45e-85) {
		tmp = y * (z / (a - t));
	} else if (t <= 9.5e+41) {
		tmp = (z * (x - y)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / t), Float64(-y), y)
	tmp = 0.0
	if (t <= -2.3e-89)
		tmp = t_1;
	elseif (t <= 1.45e-85)
		tmp = Float64(y * Float64(z / Float64(a - t)));
	elseif (t <= 9.5e+41)
		tmp = Float64(Float64(z * Float64(x - y)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * (-y) + y), $MachinePrecision]}, If[LessEqual[t, -2.3e-89], t$95$1, If[LessEqual[t, 1.45e-85], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e+41], N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.3e-89 or 9.4999999999999996e41 < t

   1. Initial program 47.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 12.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 56.5%

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

   if -2.3e-89 < t < 1.4500000000000001e-85

   1. Initial program 95.1%

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

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 59.3

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

   7. Applied rewrites 59.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 38.5%

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

   if 1.4500000000000001e-85 < t < 9.4999999999999996e41

   1. Initial program 86.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 52.2

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

   5. Applied rewrites 52.2%

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

   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 48.1%

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

Alternative 9: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.8e+53) t_1 (if (<= t 4.6e-43) (fma (/ z a) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -1.8e+53) {
		tmp = t_1;
	} else if (t <= 4.6e-43) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -1.8e+53)
		tmp = t_1;
	elseif (t <= 4.6e-43)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.8e+53], t$95$1, If[LessEqual[t, 4.6e-43], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8e53 or 4.5999999999999998e-43 < t

   1. Initial program 46.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 84.2%

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

   if -1.8e53 < t < 4.5999999999999998e-43

   1. Initial program 90.1%

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

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

      5. associate-/r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 75.5

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

   7. Applied rewrites 75.5%

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

Alternative 10: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- x y) t) y)))
   (if (<= t -1.8e+53) t_1 (if (<= t 4.6e-43) (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(z, ((x - y) / t), y);
	double tmp;
	if (t <= -1.8e+53) {
		tmp = t_1;
	} else if (t <= 4.6e-43) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(x - y) / t), y)
	tmp = 0.0
	if (t <= -1.8e+53)
		tmp = t_1;
	elseif (t <= 4.6e-43)
		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[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.8e+53], t$95$1, If[LessEqual[t, 4.6e-43], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8e53 or 4.5999999999999998e-43 < t

   1. Initial program 46.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 15.3%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 73.2%

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

   if -1.8e53 < t < 4.5999999999999998e-43

   1. Initial program 90.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 72.4

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

   5. Applied rewrites 72.4%

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

Alternative 11: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{x - y}{a}, x\right)\\ \mathbf{if}\;a \leq -1.18 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ (- x y) a) x)))
   (if (<= a -1.18e+24) t_1 (if (<= a 2.8e+139) (fma z (/ (- x y) t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, ((x - y) / a), x);
	double tmp;
	if (a <= -1.18e+24) {
		tmp = t_1;
	} else if (a <= 2.8e+139) {
		tmp = fma(z, ((x - y) / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(Float64(x - y) / a), x)
	tmp = 0.0
	if (a <= -1.18e+24)
		tmp = t_1;
	elseif (a <= 2.8e+139)
		tmp = fma(z, Float64(Float64(x - y) / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(x - y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.18e+24], t$95$1, If[LessEqual[a, 2.8e+139], N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.17999999999999997e24 or 2.7999999999999998e139 < a

   1. Initial program 69.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 53.7%

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

   if -1.17999999999999997e24 < a < 2.7999999999999998e139

   1. Initial program 65.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 19.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 68.1%

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

Alternative 12: 38.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, -y, y\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z t) (- y) y)))
   (if (<= t -4.5e+20) t_1 (if (<= t 9.5e+41) (/ (* z (- x y)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / t), -y, y);
	double tmp;
	if (t <= -4.5e+20) {
		tmp = t_1;
	} else if (t <= 9.5e+41) {
		tmp = (z * (x - y)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / t), Float64(-y), y)
	tmp = 0.0
	if (t <= -4.5e+20)
		tmp = t_1;
	elseif (t <= 9.5e+41)
		tmp = Float64(Float64(z * Float64(x - y)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * (-y) + y), $MachinePrecision]}, If[LessEqual[t, -4.5e+20], t$95$1, If[LessEqual[t, 9.5e+41], N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.5e20 or 9.4999999999999996e41 < t

   1. Initial program 44.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 50.5

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 11.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.6%

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

   if -4.5e20 < t < 9.4999999999999996e41

   1. Initial program 91.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 27.7

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

   5. Applied rewrites 27.7%

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

   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 28.1%

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

Alternative 13: 35.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* z (- x y)) t)))
   (if (<= z -5.6e+83) t_1 (if (<= z 1.62e+44) (fma a (/ y t) y) 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 <= -5.6e+83) {
		tmp = t_1;
	} else if (z <= 1.62e+44) {
		tmp = fma(a, (y / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * Float64(x - y)) / t)
	tmp = 0.0
	if (z <= -5.6e+83)
		tmp = t_1;
	elseif (z <= 1.62e+44)
		tmp = fma(a, Float64(y / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -5.6e+83], t$95$1, If[LessEqual[z, 1.62e+44], N[(a * N[(y / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.6000000000000001e83 or 1.6199999999999999e44 < z

   1. Initial program 73.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 51.9

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

   5. Applied rewrites 51.9%

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

   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 39.8%

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

   if -5.6000000000000001e83 < z < 1.6199999999999999e44

   1. Initial program 62.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 42.1%

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

Alternative 14: 32.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{z}{t}\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.68 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ z t))))
   (if (<= z -2.2e+63) t_1 (if (<= z 1.68e+44) (fma a (/ y t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (z / t);
	double tmp;
	if (z <= -2.2e+63) {
		tmp = t_1;
	} else if (z <= 1.68e+44) {
		tmp = fma(a, (y / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(z / t))
	tmp = 0.0
	if (z <= -2.2e+63)
		tmp = t_1;
	elseif (z <= 1.68e+44)
		tmp = fma(a, Float64(y / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.2e+63], t$95$1, If[LessEqual[z, 1.68e+44], N[(a * N[(y / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1999999999999999e63 or 1.68000000000000001e44 < z

   1. Initial program 73.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 49.6

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 34.7%

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

   if -2.1999999999999999e63 < z < 1.68000000000000001e44

   1. Initial program 62.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.0%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 42.8%

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

Alternative 15: 32.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{z}{t}\\ \mathbf{if}\;z \leq -3 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.76 \cdot 10^{+31}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ z t))))
   (if (<= z -3e+41) t_1 (if (<= z 1.76e+31) y t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (z / t);
	double tmp;
	if (z <= -3e+41) {
		tmp = t_1;
	} else if (z <= 1.76e+31) {
		tmp = y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z / t)
    if (z <= (-3d+41)) then
        tmp = t_1
    else if (z <= 1.76d+31) then
        tmp = y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (z / t);
	double tmp;
	if (z <= -3e+41) {
		tmp = t_1;
	} else if (z <= 1.76e+31) {
		tmp = y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x * (z / t)
	tmp = 0
	if z <= -3e+41:
		tmp = t_1
	elif z <= 1.76e+31:
		tmp = y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(z / t))
	tmp = 0.0
	if (z <= -3e+41)
		tmp = t_1;
	elseif (z <= 1.76e+31)
		tmp = y;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x * (z / t);
	tmp = 0.0;
	if (z <= -3e+41)
		tmp = t_1;
	elseif (z <= 1.76e+31)
		tmp = y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3e+41], t$95$1, If[LessEqual[z, 1.76e+31], y, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9999999999999998e41 or 1.76e31 < z

   1. Initial program 74.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 34.4%

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

   if -2.9999999999999998e41 < z < 1.76e31

   1. Initial program 61.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 60.9

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 42.5%

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

Alternative 16: 25.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := y

Derivation

1. Initial program 66.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. unsub-neg N/A

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

   13. remove-double-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 45.1

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

5. Applied rewrites 45.1%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 29.5%

   \[\leadsto y \]
9. Add Preprocessing

Developer Target 1: 87.4% 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 2024221 
(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))))