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

Percentage Accurate: 67.2% → 87.5%

Time: 11.1s

Alternatives: 19

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+94}:\\ \;\;\;\;x - \frac{x - t}{\frac{z - a}{z - y}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (fma t -1.0 x) z) (- y a) t)))
   (if (<= z -6e+120)
     t_1
     (if (<= z 6e+94) (- x (/ (- x t) (/ (- z a) (- z y)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((fma(t, -1.0, x) / z), (y - a), t);
	double tmp;
	if (z <= -6e+120) {
		tmp = t_1;
	} else if (z <= 6e+94) {
		tmp = x - ((x - t) / ((z - a) / (z - y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(fma(t, -1.0, x) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -6e+120)
		tmp = t_1;
	elseif (z <= 6e+94)
		tmp = Float64(x - Float64(Float64(x - t) / Float64(Float64(z - a) / Float64(z - y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t * -1.0 + x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -6e+120], t$95$1, If[LessEqual[z, 6e+94], N[(x - N[(N[(x - t), $MachinePrecision] / N[(N[(z - a), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6e120 or 6.0000000000000001e94 < z

   1. Initial program 38.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 90.9%

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

   if -6e120 < z < 6.0000000000000001e94

   1. Initial program 86.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lift--.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sub-neg N/A

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

      9. lift--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. un-div-inv N/A

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

      16. lower-/.f64 95.1

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

   6. Applied rewrites 95.1%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+94}:\\ \;\;\;\;x - \frac{x - t}{\frac{z - a}{z - y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\left(y - z\right) \cdot t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+96}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* (- y z) t) (- z a))))
        (t_2 (fma (/ (fma t -1.0 x) z) (- y a) t)))
   (if (<= z -4.5e+96)
     t_2
     (if (<= z -4.6e-178)
       t_1
       (if (<= z 4e-135)
         (fma (/ (- y z) a) (- t x) x)
         (if (<= z 7e+80) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (((y - z) * t) / (z - a));
	double t_2 = fma((fma(t, -1.0, x) / z), (y - a), t);
	double tmp;
	if (z <= -4.5e+96) {
		tmp = t_2;
	} else if (z <= -4.6e-178) {
		tmp = t_1;
	} else if (z <= 4e-135) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else if (z <= 7e+80) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(Float64(y - z) * t) / Float64(z - a)))
	t_2 = fma(Float64(fma(t, -1.0, x) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -4.5e+96)
		tmp = t_2;
	elseif (z <= -4.6e-178)
		tmp = t_1;
	elseif (z <= 4e-135)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	elseif (z <= 7e+80)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t * -1.0 + x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -4.5e+96], t$95$2, If[LessEqual[z, -4.6e-178], t$95$1, If[LessEqual[z, 4e-135], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 7e+80], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.49999999999999957e96 or 6.99999999999999987e80 < z

   1. Initial program 38.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 87.5%

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

   if -4.49999999999999957e96 < z < -4.59999999999999989e-178 or 4.0000000000000002e-135 < z < 6.99999999999999987e80

   1. Initial program 88.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -4.59999999999999989e-178 < z < 4.0000000000000002e-135

   1. Initial program 89.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-178}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot t}{z - a}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+80}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\left(y - z\right) \cdot t}{z - a}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* (- y z) t) (- z a)))))
   (if (<= z -4.8e+96)
     (fma (/ x z) (- y a) t)
     (if (<= z -4.6e-178)
       t_1
       (if (<= z 4e-135)
         (fma (/ (- y z) a) (- t x) x)
         (if (<= z 4.8e+83) t_1 (fma y (/ (- x t) z) t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (((y - z) * t) / (z - a));
	double tmp;
	if (z <= -4.8e+96) {
		tmp = fma((x / z), (y - a), t);
	} else if (z <= -4.6e-178) {
		tmp = t_1;
	} else if (z <= 4e-135) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else if (z <= 4.8e+83) {
		tmp = t_1;
	} else {
		tmp = fma(y, ((x - t) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(Float64(y - z) * t) / Float64(z - a)))
	tmp = 0.0
	if (z <= -4.8e+96)
		tmp = fma(Float64(x / z), Float64(y - a), t);
	elseif (z <= -4.6e-178)
		tmp = t_1;
	elseif (z <= 4e-135)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	elseif (z <= 4.8e+83)
		tmp = t_1;
	else
		tmp = fma(y, Float64(Float64(x - t) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.8e+96], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, -4.6e-178], t$95$1, If[LessEqual[z, 4e-135], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.8e+83], t$95$1, N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.79999999999999986e96

   1. Initial program 35.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 79.4%

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

   if -4.79999999999999986e96 < z < -4.59999999999999989e-178 or 4.0000000000000002e-135 < z < 4.79999999999999982e83

   1. Initial program 88.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -4.59999999999999989e-178 < z < 4.0000000000000002e-135

   1. Initial program 89.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 4.79999999999999982e83 < z

   1. Initial program 42.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-178}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot t}{z - a}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+83}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -5 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{-173}:\\ \;\;\;\;t - \frac{\left(x - t\right) \cdot \left(a - y\right)}{z}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{z - y}{z - a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -5e+50)
     t_1
     (if (<= a -5.8e-173)
       (- t (/ (* (- x t) (- a y)) z))
       (if (<= a 3e-109)
         (fma y (/ (- x t) z) t)
         (if (<= a 3.4e+28) (* (/ (- z y) (- z a)) t) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -5e+50) {
		tmp = t_1;
	} else if (a <= -5.8e-173) {
		tmp = t - (((x - t) * (a - y)) / z);
	} else if (a <= 3e-109) {
		tmp = fma(y, ((x - t) / z), t);
	} else if (a <= 3.4e+28) {
		tmp = ((z - y) / (z - a)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -5e+50)
		tmp = t_1;
	elseif (a <= -5.8e-173)
		tmp = Float64(t - Float64(Float64(Float64(x - t) * Float64(a - y)) / z));
	elseif (a <= 3e-109)
		tmp = fma(y, Float64(Float64(x - t) / z), t);
	elseif (a <= 3.4e+28)
		tmp = Float64(Float64(Float64(z - y) / Float64(z - a)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5e+50], t$95$1, If[LessEqual[a, -5.8e-173], N[(t - N[(N[(N[(x - t), $MachinePrecision] * N[(a - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3e-109], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[a, 3.4e+28], N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5e50 or 3.4e28 < a

   1. Initial program 68.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if -5e50 < a < -5.7999999999999997e-173

   1. Initial program 83.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 75.3%

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

   if -5.7999999999999997e-173 < a < 3.00000000000000021e-109

   1. Initial program 64.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 93.4%

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

   if 3.00000000000000021e-109 < a < 3.4e28

   1. Initial program 70.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.3

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. div-sub N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 73.8

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

   7. Applied rewrites 73.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{-173}:\\ \;\;\;\;t - \frac{\left(x - t\right) \cdot \left(a - y\right)}{z}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{z - y}{z - a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -5.3 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{z - y}{z - a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -5.3e+78)
     t_1
     (if (<= a 3e-109)
       (fma y (/ (- x t) z) t)
       (if (<= a 3.4e+28) (* (/ (- z y) (- z a)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -5.3e+78) {
		tmp = t_1;
	} else if (a <= 3e-109) {
		tmp = fma(y, ((x - t) / z), t);
	} else if (a <= 3.4e+28) {
		tmp = ((z - y) / (z - a)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -5.3e+78)
		tmp = t_1;
	elseif (a <= 3e-109)
		tmp = fma(y, Float64(Float64(x - t) / z), t);
	elseif (a <= 3.4e+28)
		tmp = Float64(Float64(Float64(z - y) / Float64(z - a)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.3e+78], t$95$1, If[LessEqual[a, 3e-109], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[a, 3.4e+28], N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.29999999999999961e78 or 3.4e28 < a

   1. Initial program 68.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if -5.29999999999999961e78 < a < 3.00000000000000021e-109

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 80.6%

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

   if 3.00000000000000021e-109 < a < 3.4e28

   1. Initial program 70.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.3

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. div-sub N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 73.8

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

   7. Applied rewrites 73.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{z - y}{z - a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -5.3 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{z - y}{z - a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ y a) x)))
   (if (<= a -5.3e+78)
     t_1
     (if (<= a 3e-109)
       (fma y (/ (- x t) z) t)
       (if (<= a 3.4e+28) (* (/ (- z y) (- z a)) t) 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 <= -5.3e+78) {
		tmp = t_1;
	} else if (a <= 3e-109) {
		tmp = fma(y, ((x - t) / z), t);
	} else if (a <= 3.4e+28) {
		tmp = ((z - y) / (z - a)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(y / a), x)
	tmp = 0.0
	if (a <= -5.3e+78)
		tmp = t_1;
	elseif (a <= 3e-109)
		tmp = fma(y, Float64(Float64(x - t) / z), t);
	elseif (a <= 3.4e+28)
		tmp = Float64(Float64(Float64(z - y) / Float64(z - a)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.3e+78], t$95$1, If[LessEqual[a, 3e-109], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[a, 3.4e+28], N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.29999999999999961e78 or 3.4e28 < a

   1. Initial program 68.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 72.6

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

   7. Applied rewrites 72.6%

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

   if -5.29999999999999961e78 < a < 3.00000000000000021e-109

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 80.6%

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

   if 3.00000000000000021e-109 < a < 3.4e28

   1. Initial program 70.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.3

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. div-sub N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 73.8

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

   7. Applied rewrites 73.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{z - y}{z - a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -5.3 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.45 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;a \leq 3.35 \cdot 10^{+28}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ y a) x)))
   (if (<= a -5.3e+78)
     t_1
     (if (<= a 2.45e-107)
       (fma y (/ (- x t) z) t)
       (if (<= a 3.35e+28) (* (- z y) (/ t (- z a))) 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 <= -5.3e+78) {
		tmp = t_1;
	} else if (a <= 2.45e-107) {
		tmp = fma(y, ((x - t) / z), t);
	} else if (a <= 3.35e+28) {
		tmp = (z - y) * (t / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(y / a), x)
	tmp = 0.0
	if (a <= -5.3e+78)
		tmp = t_1;
	elseif (a <= 2.45e-107)
		tmp = fma(y, Float64(Float64(x - t) / z), t);
	elseif (a <= 3.35e+28)
		tmp = Float64(Float64(z - y) * Float64(t / Float64(z - a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.3e+78], t$95$1, If[LessEqual[a, 2.45e-107], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[a, 3.35e+28], N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.29999999999999961e78 or 3.35e28 < a

   1. Initial program 68.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 72.6

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

   7. Applied rewrites 72.6%

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

   if -5.29999999999999961e78 < a < 2.4499999999999999e-107

   1. Initial program 72.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 80.9%

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

   if 2.4499999999999999e-107 < a < 3.35e28

   1. Initial program 71.2%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 67.8

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

   5. Applied rewrites 67.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq 2.45 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;a \leq 3.35 \cdot 10^{+28}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{z - y}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (fma t -1.0 x) z) (- y a) t)))
   (if (<= z -6e+120)
     t_1
     (if (<= z 6e+94) (fma (- t x) (/ (- z y) (- z a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((fma(t, -1.0, x) / z), (y - a), t);
	double tmp;
	if (z <= -6e+120) {
		tmp = t_1;
	} else if (z <= 6e+94) {
		tmp = fma((t - x), ((z - y) / (z - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(fma(t, -1.0, x) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -6e+120)
		tmp = t_1;
	elseif (z <= 6e+94)
		tmp = fma(Float64(t - x), Float64(Float64(z - y) / Float64(z - a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t * -1.0 + x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -6e+120], t$95$1, If[LessEqual[z, 6e+94], N[(N[(t - x), $MachinePrecision] * N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6e120 or 6.0000000000000001e94 < z

   1. Initial program 38.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 90.9%

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

   if -6e120 < z < 6.0000000000000001e94

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{z - y}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t, -1, x\right)}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq -1.56 \cdot 10^{-270}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.2e+87)
   (fma (- x) (/ y a) x)
   (if (<= a -1.56e-270)
     (* (/ (- z y) z) t)
     (if (<= a 6.5e+40) (fma y (/ x z) t) (fma y (/ t a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.2e+87) {
		tmp = fma(-x, (y / a), x);
	} else if (a <= -1.56e-270) {
		tmp = ((z - y) / z) * t;
	} else if (a <= 6.5e+40) {
		tmp = fma(y, (x / z), t);
	} else {
		tmp = fma(y, (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.2e+87)
		tmp = fma(Float64(-x), Float64(y / a), x);
	elseif (a <= -1.56e-270)
		tmp = Float64(Float64(Float64(z - y) / z) * t);
	elseif (a <= 6.5e+40)
		tmp = fma(y, Float64(x / z), t);
	else
		tmp = fma(y, Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.2e+87], N[((-x) * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, -1.56e-270], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[a, 6.5e+40], N[(y * N[(x / z), $MachinePrecision] + t), $MachinePrecision], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.2000000000000001e87

   1. Initial program 70.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.3%

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

   if -2.2000000000000001e87 < a < -1.55999999999999999e-270

   1. Initial program 75.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.0%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 63.2%

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

   if -1.55999999999999999e-270 < a < 6.5000000000000001e40

   1. Initial program 70.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.5%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 61.8%

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

   if 6.5000000000000001e40 < a

   1. Initial program 65.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 63.0%

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

Alternative 10: 36.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+119}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-176}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+80}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e+119)
   t
   (if (<= z -4.8e-176)
     (* 1.0 x)
     (if (<= z 9.8e-38) (* (/ y a) t) (if (<= z 6.5e+80) (* 1.0 x) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e+119) {
		tmp = t;
	} else if (z <= -4.8e-176) {
		tmp = 1.0 * x;
	} else if (z <= 9.8e-38) {
		tmp = (y / a) * t;
	} else if (z <= 6.5e+80) {
		tmp = 1.0 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.2d+119)) then
        tmp = t
    else if (z <= (-4.8d-176)) then
        tmp = 1.0d0 * x
    else if (z <= 9.8d-38) then
        tmp = (y / a) * t
    else if (z <= 6.5d+80) then
        tmp = 1.0d0 * x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e+119) {
		tmp = t;
	} else if (z <= -4.8e-176) {
		tmp = 1.0 * x;
	} else if (z <= 9.8e-38) {
		tmp = (y / a) * t;
	} else if (z <= 6.5e+80) {
		tmp = 1.0 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.2e+119:
		tmp = t
	elif z <= -4.8e-176:
		tmp = 1.0 * x
	elif z <= 9.8e-38:
		tmp = (y / a) * t
	elif z <= 6.5e+80:
		tmp = 1.0 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e+119)
		tmp = t;
	elseif (z <= -4.8e-176)
		tmp = Float64(1.0 * x);
	elseif (z <= 9.8e-38)
		tmp = Float64(Float64(y / a) * t);
	elseif (z <= 6.5e+80)
		tmp = Float64(1.0 * x);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.2e+119)
		tmp = t;
	elseif (z <= -4.8e-176)
		tmp = 1.0 * x;
	elseif (z <= 9.8e-38)
		tmp = (y / a) * t;
	elseif (z <= 6.5e+80)
		tmp = 1.0 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e+119], t, If[LessEqual[z, -4.8e-176], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 9.8e-38], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 6.5e+80], N[(1.0 * x), $MachinePrecision], t]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2000000000000001e119 or 6.4999999999999998e80 < z

   1. Initial program 39.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 38.4

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

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot x + t \]

      4. mul0-lft N/A

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

      5. lower-+.f64 54.9

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

   7. Applied rewrites 54.9%

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

   if -2.2000000000000001e119 < z < -4.80000000000000012e-176 or 9.80000000000000078e-38 < z < 6.4999999999999998e80

   1. Initial program 82.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. times-frac N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in a around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 34.6%

      \[\leadsto 1 \cdot x \]

   if -4.80000000000000012e-176 < z < 9.80000000000000078e-38

   1. Initial program 91.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. div-sub N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 51.0

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

   7. Applied rewrites 51.0%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 45.0%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+119}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-176}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+80}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -5.3 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3700000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ y a) x)))
   (if (<= a -5.3e+78)
     t_1
     (if (<= a 3700000000000.0) (fma y (/ (- x t) z) t) 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 <= -5.3e+78) {
		tmp = t_1;
	} else if (a <= 3700000000000.0) {
		tmp = fma(y, ((x - t) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(y / a), x)
	tmp = 0.0
	if (a <= -5.3e+78)
		tmp = t_1;
	elseif (a <= 3700000000000.0)
		tmp = fma(y, Float64(Float64(x - t) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.3e+78], t$95$1, If[LessEqual[a, 3700000000000.0], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.29999999999999961e78 or 3.7e12 < a

   1. Initial program 69.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 72.4

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

   7. Applied rewrites 72.4%

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

   if -5.29999999999999961e78 < a < 3.7e12

   1. Initial program 71.2%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.7%

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

Alternative 12: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{if}\;a \leq -5.4 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3700000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) a) y x)))
   (if (<= a -5.4e+78)
     t_1
     (if (<= a 3700000000000.0) (fma y (/ (- x t) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / a), y, x);
	double tmp;
	if (a <= -5.4e+78) {
		tmp = t_1;
	} else if (a <= 3700000000000.0) {
		tmp = fma(y, ((x - t) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / a), y, x)
	tmp = 0.0
	if (a <= -5.4e+78)
		tmp = t_1;
	elseif (a <= 3700000000000.0)
		tmp = fma(y, Float64(Float64(x - t) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -5.4e+78], t$95$1, If[LessEqual[a, 3700000000000.0], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.40000000000000009e78 or 3.7e12 < a

   1. Initial program 69.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -5.40000000000000009e78 < a < 3.7e12

   1. Initial program 71.2%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.7%

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

Alternative 13: 63.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -6.6e+78)
   (fma (- x) (/ y a) x)
   (if (<= a 7.4e+40) (fma y (/ (- x t) z) t) (fma y (/ t a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -6.6e+78) {
		tmp = fma(-x, (y / a), x);
	} else if (a <= 7.4e+40) {
		tmp = fma(y, ((x - t) / z), t);
	} else {
		tmp = fma(y, (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -6.6e+78)
		tmp = fma(Float64(-x), Float64(y / a), x);
	elseif (a <= 7.4e+40)
		tmp = fma(y, Float64(Float64(x - t) / z), t);
	else
		tmp = fma(y, Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -6.6e+78], N[((-x) * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 7.4e+40], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.6e78

   1. Initial program 69.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.8%

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

   if -6.6e78 < a < 7.4e40

   1. Initial program 72.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.0%

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

   if 7.4e40 < a

   1. Initial program 65.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 63.0%

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

Alternative 14: 56.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.9e+120)
   (fma a (/ (- t x) z) t)
   (if (<= z 9e+82) (fma y (/ t a) x) (fma y (/ x z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e+120) {
		tmp = fma(a, ((t - x) / z), t);
	} else if (z <= 9e+82) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = fma(y, (x / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.9e+120)
		tmp = fma(a, Float64(Float64(t - x) / z), t);
	elseif (z <= 9e+82)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = fma(y, Float64(x / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.9e+120], N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 9e+82], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8999999999999999e120

   1. Initial program 37.2%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 74.4%

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

   if -1.8999999999999999e120 < z < 8.9999999999999993e82

   1. Initial program 86.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 66.0

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

   7. Applied rewrites 66.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 55.2%

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

   if 8.9999999999999993e82 < z

   1. Initial program 42.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.3%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 72.2%

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

Alternative 15: 56.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{x}{z}, t\right)\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ x z) t)))
   (if (<= z -1.12e+120) t_1 (if (<= z 9e+82) (fma y (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (x / z), t);
	double tmp;
	if (z <= -1.12e+120) {
		tmp = t_1;
	} else if (z <= 9e+82) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(x / z), t)
	tmp = 0.0
	if (z <= -1.12e+120)
		tmp = t_1;
	elseif (z <= 9e+82)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(x / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.12e+120], t$95$1, If[LessEqual[z, 9e+82], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.12000000000000005e120 or 8.9999999999999993e82 < z

   1. Initial program 39.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 70.0%

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

   if -1.12000000000000005e120 < z < 8.9999999999999993e82

   1. Initial program 86.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 66.0

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

   7. Applied rewrites 66.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 55.2%

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

Alternative 16: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{+101}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.6e+101)
   (* 1.0 x)
   (if (<= a 3.7e+43) (fma y (/ x z) t) (* 1.0 x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.6e+101) {
		tmp = 1.0 * x;
	} else if (a <= 3.7e+43) {
		tmp = fma(y, (x / z), t);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.6e+101)
		tmp = Float64(1.0 * x);
	elseif (a <= 3.7e+43)
		tmp = fma(y, Float64(x / z), t);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.6e+101], N[(1.0 * x), $MachinePrecision], If[LessEqual[a, 3.7e+43], N[(y * N[(x / z), $MachinePrecision] + t), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.60000000000000003e101 or 3.7000000000000001e43 < a

   1. Initial program 67.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. times-frac N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in a around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 48.8%

      \[\leadsto 1 \cdot x \]

   if -1.60000000000000003e101 < a < 3.7000000000000001e43

   1. Initial program 72.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 70.5%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 55.7%

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

Alternative 17: 38.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{+100}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;a \leq 3.65 \cdot 10^{+43}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -7.6e+100) (* 1.0 x) (if (<= a 3.65e+43) t (* 1.0 x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7.6e+100) {
		tmp = 1.0 * x;
	} else if (a <= 3.65e+43) {
		tmp = t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-7.6d+100)) then
        tmp = 1.0d0 * x
    else if (a <= 3.65d+43) then
        tmp = t
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7.6e+100) {
		tmp = 1.0 * x;
	} else if (a <= 3.65e+43) {
		tmp = t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -7.6e+100:
		tmp = 1.0 * x
	elif a <= 3.65e+43:
		tmp = t
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -7.6e+100)
		tmp = Float64(1.0 * x);
	elseif (a <= 3.65e+43)
		tmp = t;
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -7.6e+100)
		tmp = 1.0 * x;
	elseif (a <= 3.65e+43)
		tmp = t;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -7.6e+100], N[(1.0 * x), $MachinePrecision], If[LessEqual[a, 3.65e+43], t, N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.59999999999999927e100 or 3.6499999999999998e43 < a

   1. Initial program 67.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. times-frac N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in a around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 48.8%

      \[\leadsto 1 \cdot x \]

   if -7.59999999999999927e100 < a < 3.6499999999999998e43

   1. Initial program 72.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - z\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(y - z\right) \cdot t}}{a - z} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y - z, \mathsf{neg}\left(x\right), \left(y - z\right) \cdot t\right)}}{a - z} \]

      7. lower-neg.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(y - z, \color{blue}{-x}, \left(y - z\right) \cdot t\right)}{a - z} \]

      8. lower-*.f64 70.1

         \[\leadsto x + \frac{\mathsf{fma}\left(y - z, -x, \color{blue}{\left(y - z\right) \cdot t}\right)}{a - z} \]

   4. Applied rewrites 70.1%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y - z, -x, \left(y - z\right) \cdot t\right)}}{a - z} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t + \left(x + -1 \cdot x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot x\right) + t} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot x} + t \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot x + t \]

      4. mul0-lft N/A

         \[\leadsto \color{blue}{0} + t \]

      5. lower-+.f64 33.8

         \[\leadsto \color{blue}{0 + t} \]

   7. Applied rewrites 33.8%

      \[\leadsto \color{blue}{0 + t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{+100}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;a \leq 3.65 \cdot 10^{+43}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 24.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 t)

C

double code(double x, double y, double z, double t, double a) {
	return 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 = t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return t;
}

Python

def code(x, y, z, t, a):
	return t

Julia

function code(x, y, z, t, a)
	return t
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = t;
end

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 70.4%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x + \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} \]

   2. lift--.f64 N/A

      \[\leadsto x + \frac{\left(y - z\right) \cdot \color{blue}{\left(t - x\right)}}{a - z} \]

   3. sub-neg N/A

      \[\leadsto x + \frac{\left(y - z\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}}{a - z} \]

   4. +-commutative N/A

      \[\leadsto x + \frac{\left(y - z\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + t\right)}}{a - z} \]

   5. distribute-lft-in N/A

      \[\leadsto x + \frac{\color{blue}{\left(y - z\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(y - z\right) \cdot t}}{a - z} \]

   6. lower-fma.f64 N/A

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y - z, \mathsf{neg}\left(x\right), \left(y - z\right) \cdot t\right)}}{a - z} \]

   7. lower-neg.f64 N/A

      \[\leadsto x + \frac{\mathsf{fma}\left(y - z, \color{blue}{-x}, \left(y - z\right) \cdot t\right)}{a - z} \]

   8. lower-*.f64 69.2

      \[\leadsto x + \frac{\mathsf{fma}\left(y - z, -x, \color{blue}{\left(y - z\right) \cdot t}\right)}{a - z} \]

4. Applied rewrites 69.2%

   \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y - z, -x, \left(y - z\right) \cdot t\right)}}{a - z} \]

5. Taylor expanded in z around inf

   \[\leadsto \color{blue}{t + \left(x + -1 \cdot x\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(x + -1 \cdot x\right) + t} \]

   2. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot x} + t \]

   3. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot x + t \]

   4. mul0-lft N/A

      \[\leadsto \color{blue}{0} + t \]

   5. lower-+.f64 26.4

      \[\leadsto \color{blue}{0 + t} \]

7. Applied rewrites 26.4%

   \[\leadsto \color{blue}{0 + t} \]

8. Final simplification 26.4%

   \[\leadsto t \]
9. Add Preprocessing

Alternative 19: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 70.4%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot \left(y - z\right)}{a - z}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \left(y - z\right)}{a - z} + x} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \left(y - z\right)}{a - z}\right)\right)} + x \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + x \]

   4. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + x \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + x \]

   6. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

   11. distribute-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

   12. unsub-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

   13. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

   14. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

   16. lower--.f64 38.1

       \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

5. Applied rewrites 38.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

6. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{-1 \cdot x} \]
7. Step-by-step derivation

8. Applied rewrites 2.7%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer Target 1: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ y z) (- t x)))))
   (if (< z -1.2536131056095036e+188)
     t_1
     (if (< z 4.446702369113811e+64)
       (+ x (/ (- y z) (/ (- a z) (- t x))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - 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 = t - ((y / z) * (t - x))
    if (z < (-1.2536131056095036d+188)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x + ((y - z) / ((a - z) / (t - 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 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - ((y / z) * (t - x))
	tmp = 0
	if z < -1.2536131056095036e+188:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x + ((y - z) / ((a - z) / (t - x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y / z) * Float64(t - x)))
	tmp = 0.0
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x + Float64(Float64(y - z) / Float64(Float64(a - z) / Float64(t - x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - ((y / z) * (t - x));
	tmp = 0.0;
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y / z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.2536131056095036e+188], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x + N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -125361310560950360000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- t (* (/ y z) (- t x))) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x))))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))