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

Percentage Accurate: 67.7% → 88.9%

Time: 12.9s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -1.36 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) (- z a) y)))
   (if (<= t -1.36e+141)
     t_1
     (if (<= t 9e+107) (fma (/ (- z t) (- a t)) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), (z - a), y);
	double tmp;
	if (t <= -1.36e+141) {
		tmp = t_1;
	} else if (t <= 9e+107) {
		tmp = fma(((z - t) / (a - t)), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - y) / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -1.36e+141)
		tmp = t_1;
	elseif (t <= 9e+107)
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.36e+141], t$95$1, If[LessEqual[t, 9e+107], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.36e141 or 9e107 < t

   1. Initial program 33.5%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\frac{y - x}{t}\right)\right), \color{blue}{\left(z - a\right)}, y\right) \]

   5. Simplified 89.3%

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

   if -1.36e141 < t < 9e107

   1. Initial program 83.1%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{z - t}{a - t}\right), \color{blue}{\left(y - x\right)}, x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(z - t\right), \left(a - t\right)\right), \left(\color{blue}{y} - x\right), x\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, t\right), \left(a - t\right)\right), \left(y - x\right), x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, t\right), \mathsf{\_.f64}\left(a, t\right)\right), \left(y - x\right), x\right) \]

      8. --lowering--.f64 92.5%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, t\right), \mathsf{\_.f64}\left(a, t\right)\right), \mathsf{\_.f64}\left(y, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 92.5%

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

Alternative 2: 62.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+136}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;t \leq -7.6 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{x}{a - t}, x\right)\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-76}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z}{t} + -1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.85e+136)
   (* y (- 1.0 (/ z t)))
   (if (<= t -7.6e+104)
     (fma (- t z) (/ x (- a t)) x)
     (if (<= t -1.45e-76)
       (/ (* y (- z t)) (- a t))
       (if (<= t 2.5e-55)
         (fma z (/ (- y x) a) x)
         (if (<= t 1.8e+118)
           (fma (- x y) (+ (/ z t) -1.0) x)
           (fma a (/ (- y x) t) y)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.85e+136) {
		tmp = y * (1.0 - (z / t));
	} else if (t <= -7.6e+104) {
		tmp = fma((t - z), (x / (a - t)), x);
	} else if (t <= -1.45e-76) {
		tmp = (y * (z - t)) / (a - t);
	} else if (t <= 2.5e-55) {
		tmp = fma(z, ((y - x) / a), x);
	} else if (t <= 1.8e+118) {
		tmp = fma((x - y), ((z / t) + -1.0), x);
	} else {
		tmp = fma(a, ((y - x) / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.85e+136)
		tmp = Float64(y * Float64(1.0 - Float64(z / t)));
	elseif (t <= -7.6e+104)
		tmp = fma(Float64(t - z), Float64(x / Float64(a - t)), x);
	elseif (t <= -1.45e-76)
		tmp = Float64(Float64(y * Float64(z - t)) / Float64(a - t));
	elseif (t <= 2.5e-55)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	elseif (t <= 1.8e+118)
		tmp = fma(Float64(x - y), Float64(Float64(z / t) + -1.0), x);
	else
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.85e+136], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.6e+104], N[(N[(t - z), $MachinePrecision] * N[(x / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, -1.45e-76], N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e-55], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.8e+118], N[(N[(x - y), $MachinePrecision] * N[(N[(z / t), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.85000000000000005e136

   1. Initial program 27.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{z - t}{t}\right)}, x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      12. div-sub N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} - \color{blue}{\frac{t}{t}}\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{t}{t}\right)\right)}\right), x\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + -1\right), x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + \color{blue}{\frac{z}{t}}\right), x\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{z}{t}\right)}\right), x\right) \]

      18. /-lowering-/.f64 52.2%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 52.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(\frac{z}{t}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 - \color{blue}{\frac{z}{t}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{z}{t}\right)}\right)\right) \]

      13. /-lowering-/.f64 76.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right)\right) \]

   8. Simplified 76.4%

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

   if -1.85000000000000005e136 < t < -7.59999999999999938e104

   1. Initial program 45.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(z - t\right)\right), \color{blue}{\left(\frac{x}{a - t}\right)}, x\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(z - t\right)\right)\right), \left(\frac{\color{blue}{x}}{a - t}\right), x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\frac{x}{a - t}\right), x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)\right)\right), \left(\frac{x}{a - t}\right), x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(z\right)\right)\right), \left(\frac{\color{blue}{x}}{a - t}\right), x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - z\right), \left(\frac{\color{blue}{x}}{a - t}\right), x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(t - z\right), \left(\frac{x}{a - t}\right), x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(t, z\right), \left(\frac{\color{blue}{x}}{a - t}\right), x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(t, z\right), \mathsf{/.f64}\left(x, \color{blue}{\left(a - t\right)}\right), x\right) \]

      21. --lowering--.f64 99.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(t, z\right), \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 99.8%

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

   if -7.59999999999999938e104 < t < -1.4500000000000001e-76

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(z - t\right)\right), \color{blue}{\left(a - t\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z - t\right)\right), \left(\color{blue}{a} - t\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, t\right)\right), \left(a - t\right)\right) \]

      4. --lowering--.f64 61.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, t\right)\right), \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right) \]

   5. Simplified 61.7%

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

   if -1.4500000000000001e-76 < t < 2.5000000000000001e-55

   1. Initial program 91.5%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \color{blue}{\left(\frac{y - x}{a}\right)}, x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{a}\right), x\right) \]

      5. --lowering--.f64 82.7%

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), a\right), x\right) \]

   5. Simplified 82.7%

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

   if 2.5000000000000001e-55 < t < 1.8e118

   1. Initial program 75.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{z - t}{t}\right)}, x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      12. div-sub N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} - \color{blue}{\frac{t}{t}}\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{t}{t}\right)\right)}\right), x\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + -1\right), x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + \color{blue}{\frac{z}{t}}\right), x\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{z}{t}\right)}\right), x\right) \]

      18. /-lowering-/.f64 50.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 50.8%

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

   if 1.8e118 < t

   1. Initial program 35.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{t}{a - t}\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{t}{a - t}\right), x\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \color{blue}{\left(a - t\right)}\right), x\right) \]

      16. --lowering--.f64 44.9%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 44.9%

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

   6. Taylor expanded in t around -inf

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

      1. associate-+r+ N/A

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

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

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. distribute-rgt1-in N/A

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

      14. metadata-eval N/A

          \[\leadsto \left(0 \cdot x + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      15. mul0-lft N/A

          \[\leadsto \left(0 + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      16. *-lft-identity N/A

          \[\leadsto \left(0 + 1 \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      17. metadata-eval N/A

          \[\leadsto \left(0 + \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      18. cancel-sign-sub-inv N/A

          \[\leadsto \left(0 - -1 \cdot y\right) + \color{blue}{-1} \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      19. neg-sub0 N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

      22. associate-*r/ N/A

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

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+136}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;t \leq -7.6 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{x}{a - t}, x\right)\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-76}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z}{t} + -1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;t \leq -6.8 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{x}{a - t}, x\right)\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -6.5e+134)
   (* y (- 1.0 (/ z t)))
   (if (<= t -6.8e+104)
     (fma (- t z) (/ x (- a t)) x)
     (if (<= t -4.2e-77)
       (/ (* y (- z t)) (- a t))
       (if (<= t 2.8e+88) (fma z (/ (- y x) a) x) (fma a (/ (- y x) t) y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -6.5e+134) {
		tmp = y * (1.0 - (z / t));
	} else if (t <= -6.8e+104) {
		tmp = fma((t - z), (x / (a - t)), x);
	} else if (t <= -4.2e-77) {
		tmp = (y * (z - t)) / (a - t);
	} else if (t <= 2.8e+88) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = fma(a, ((y - x) / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -6.5e+134)
		tmp = Float64(y * Float64(1.0 - Float64(z / t)));
	elseif (t <= -6.8e+104)
		tmp = fma(Float64(t - z), Float64(x / Float64(a - t)), x);
	elseif (t <= -4.2e-77)
		tmp = Float64(Float64(y * Float64(z - t)) / Float64(a - t));
	elseif (t <= 2.8e+88)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -6.5e+134], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6.8e+104], N[(N[(t - z), $MachinePrecision] * N[(x / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, -4.2e-77], N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+88], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6.5e134

   1. Initial program 27.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{z - t}{t}\right)}, x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      12. div-sub N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} - \color{blue}{\frac{t}{t}}\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{t}{t}\right)\right)}\right), x\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + -1\right), x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + \color{blue}{\frac{z}{t}}\right), x\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{z}{t}\right)}\right), x\right) \]

      18. /-lowering-/.f64 52.2%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 52.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(\frac{z}{t}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 - \color{blue}{\frac{z}{t}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{z}{t}\right)}\right)\right) \]

      13. /-lowering-/.f64 76.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right)\right) \]

   8. Simplified 76.4%

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

   if -6.5e134 < t < -6.7999999999999994e104

   1. Initial program 45.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(z - t\right)\right), \color{blue}{\left(\frac{x}{a - t}\right)}, x\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(z - t\right)\right)\right), \left(\frac{\color{blue}{x}}{a - t}\right), x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), \left(\frac{x}{a - t}\right), x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)\right)\right), \left(\frac{x}{a - t}\right), x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(z\right)\right)\right), \left(\frac{\color{blue}{x}}{a - t}\right), x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - z\right), \left(\frac{\color{blue}{x}}{a - t}\right), x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(t - z\right), \left(\frac{x}{a - t}\right), x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(t, z\right), \left(\frac{\color{blue}{x}}{a - t}\right), x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(t, z\right), \mathsf{/.f64}\left(x, \color{blue}{\left(a - t\right)}\right), x\right) \]

      21. --lowering--.f64 99.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(t, z\right), \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 99.8%

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

   if -6.7999999999999994e104 < t < -4.20000000000000031e-77

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(z - t\right)\right), \color{blue}{\left(a - t\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z - t\right)\right), \left(\color{blue}{a} - t\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, t\right)\right), \left(a - t\right)\right) \]

      4. --lowering--.f64 61.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, t\right)\right), \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right) \]

   5. Simplified 61.7%

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

   if -4.20000000000000031e-77 < t < 2.79999999999999989e88

   1. Initial program 87.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \color{blue}{\left(\frac{y - x}{a}\right)}, x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{a}\right), x\right) \]

      5. --lowering--.f64 70.6%

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), a\right), x\right) \]

   5. Simplified 70.6%

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

   if 2.79999999999999989e88 < t

   1. Initial program 42.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{t}{a - t}\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{t}{a - t}\right), x\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \color{blue}{\left(a - t\right)}\right), x\right) \]

      16. --lowering--.f64 46.1%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 46.1%

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

   6. Taylor expanded in t around -inf

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

      1. associate-+r+ N/A

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

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

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. distribute-rgt1-in N/A

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

      14. metadata-eval N/A

          \[\leadsto \left(0 \cdot x + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      15. mul0-lft N/A

          \[\leadsto \left(0 + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      16. *-lft-identity N/A

          \[\leadsto \left(0 + 1 \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      17. metadata-eval N/A

          \[\leadsto \left(0 + \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      18. cancel-sign-sub-inv N/A

          \[\leadsto \left(0 - -1 \cdot y\right) + \color{blue}{-1} \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      19. neg-sub0 N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

      22. associate-*r/ N/A

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

   8. Simplified 63.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 61.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.7 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{+107}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -6.7e+134)
   (* y (- 1.0 (/ z t)))
   (if (<= t -8.2e+107)
     (+ x (/ (* y z) a))
     (if (<= t -7.5e-77)
       (/ (* y (- z t)) (- a t))
       (if (<= t 3.4e+87) (fma z (/ (- y x) a) x) (fma a (/ (- y x) t) y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -6.7e+134) {
		tmp = y * (1.0 - (z / t));
	} else if (t <= -8.2e+107) {
		tmp = x + ((y * z) / a);
	} else if (t <= -7.5e-77) {
		tmp = (y * (z - t)) / (a - t);
	} else if (t <= 3.4e+87) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = fma(a, ((y - x) / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -6.7e+134)
		tmp = Float64(y * Float64(1.0 - Float64(z / t)));
	elseif (t <= -8.2e+107)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	elseif (t <= -7.5e-77)
		tmp = Float64(Float64(y * Float64(z - t)) / Float64(a - t));
	elseif (t <= 3.4e+87)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -6.7e+134], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -8.2e+107], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.5e-77], N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e+87], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6.6999999999999997e134

   1. Initial program 27.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{z - t}{t}\right)}, x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      12. div-sub N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} - \color{blue}{\frac{t}{t}}\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{t}{t}\right)\right)}\right), x\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + -1\right), x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + \color{blue}{\frac{z}{t}}\right), x\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{z}{t}\right)}\right), x\right) \]

      18. /-lowering-/.f64 52.2%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 52.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(\frac{z}{t}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 - \color{blue}{\frac{z}{t}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{z}{t}\right)}\right)\right) \]

      13. /-lowering-/.f64 76.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right)\right) \]

   8. Simplified 76.4%

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

   if -6.6999999999999997e134 < t < -8.1999999999999998e107

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{a}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(z \cdot \left(y - x\right)\right), \color{blue}{a}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(y - x\right)\right), a\right)\right) \]

      3. --lowering--.f64 35.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, x\right)\right), a\right)\right) \]

   5. Simplified 35.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{y}\right), a\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.4%

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

   if -8.1999999999999998e107 < t < -7.5000000000000006e-77

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(z - t\right)\right), \color{blue}{\left(a - t\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z - t\right)\right), \left(\color{blue}{a} - t\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, t\right)\right), \left(a - t\right)\right) \]

      4. --lowering--.f64 60.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(z, t\right)\right), \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right) \]

   5. Simplified 60.1%

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

   if -7.5000000000000006e-77 < t < 3.4000000000000002e87

   1. Initial program 87.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \color{blue}{\left(\frac{y - x}{a}\right)}, x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{a}\right), x\right) \]

      5. --lowering--.f64 70.6%

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), a\right), x\right) \]

   5. Simplified 70.6%

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

   if 3.4000000000000002e87 < t

   1. Initial program 42.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{t}{a - t}\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{t}{a - t}\right), x\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \color{blue}{\left(a - t\right)}\right), x\right) \]

      16. --lowering--.f64 46.1%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 46.1%

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

   6. Taylor expanded in t around -inf

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

      1. associate-+r+ N/A

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

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

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. distribute-rgt1-in N/A

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

      14. metadata-eval N/A

          \[\leadsto \left(0 \cdot x + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      15. mul0-lft N/A

          \[\leadsto \left(0 + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      16. *-lft-identity N/A

          \[\leadsto \left(0 + 1 \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      17. metadata-eval N/A

          \[\leadsto \left(0 + \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      18. cancel-sign-sub-inv N/A

          \[\leadsto \left(0 - -1 \cdot y\right) + \color{blue}{-1} \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      19. neg-sub0 N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

      22. associate-*r/ N/A

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

   8. Simplified 63.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.7 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{+107}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-208}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) (- z a) y)))
   (if (<= t -1.4e+133)
     t_1
     (if (<= t -5.6e-208)
       (fma (/ (- z t) (- a t)) y x)
       (if (<= t 1.6e-67) (fma z (/ (- y x) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), (z - a), y);
	double tmp;
	if (t <= -1.4e+133) {
		tmp = t_1;
	} else if (t <= -5.6e-208) {
		tmp = fma(((z - t) / (a - t)), y, x);
	} else if (t <= 1.6e-67) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - y) / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -1.4e+133)
		tmp = t_1;
	elseif (t <= -5.6e-208)
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), y, x);
	elseif (t <= 1.6e-67)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.4e+133], t$95$1, If[LessEqual[t, -5.6e-208], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 1.6e-67], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.40000000000000008e133 or 1.60000000000000011e-67 < t

   1. Initial program 46.0%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\frac{y - x}{t}\right)\right), \color{blue}{\left(z - a\right)}, y\right) \]

   5. Simplified 80.9%

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

   if -1.40000000000000008e133 < t < -5.60000000000000003e-208

   1. Initial program 80.9%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{z - t}{a - t}\right), \color{blue}{\left(y - x\right)}, x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(z - t\right), \left(a - t\right)\right), \left(\color{blue}{y} - x\right), x\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, t\right), \left(a - t\right)\right), \left(y - x\right), x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, t\right), \mathsf{\_.f64}\left(a, t\right)\right), \left(y - x\right), x\right) \]

      8. --lowering--.f64 90.5%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, t\right), \mathsf{\_.f64}\left(a, t\right)\right), \mathsf{\_.f64}\left(y, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 90.5%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, t\right), \mathsf{\_.f64}\left(a, t\right)\right), \color{blue}{y}, x\right) \]
   6. Step-by-step derivation

   7. Simplified 75.8%

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

   if -5.60000000000000003e-208 < t < 1.60000000000000011e-67

   1. Initial program 92.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \color{blue}{\left(\frac{y - x}{a}\right)}, x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{a}\right), x\right) \]

      5. --lowering--.f64 89.5%

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), a\right), x\right) \]

   5. Simplified 89.5%

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

Alternative 6: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -3 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) (- z a) y)))
   (if (<= t -3e+54) t_1 (if (<= t 5e-66) (fma z (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), (z - a), y);
	double tmp;
	if (t <= -3e+54) {
		tmp = t_1;
	} else if (t <= 5e-66) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - y) / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -3e+54)
		tmp = t_1;
	elseif (t <= 5e-66)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -3e+54], t$95$1, If[LessEqual[t, 5e-66], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9999999999999999e54 or 4.99999999999999962e-66 < t

   1. Initial program 47.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\frac{y - x}{t}\right)\right), \color{blue}{\left(z - a\right)}, y\right) \]

   5. Simplified 78.6%

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

   if -2.9999999999999999e54 < t < 4.99999999999999962e-66

   1. Initial program 90.7%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \color{blue}{\left(\frac{y - x}{a}\right)}, x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{a}\right), x\right) \]

      5. --lowering--.f64 75.5%

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), a\right), x\right) \]

   5. Simplified 75.5%

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

Alternative 7: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -6.5e+134)
   (* y (- 1.0 (/ z t)))
   (if (<= t 6.5e+87) (fma z (/ (- y x) a) x) (fma a (/ (- y x) t) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -6.5e+134) {
		tmp = y * (1.0 - (z / t));
	} else if (t <= 6.5e+87) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = fma(a, ((y - x) / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -6.5e+134)
		tmp = Float64(y * Float64(1.0 - Float64(z / t)));
	elseif (t <= 6.5e+87)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -6.5e+134], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e+87], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.5e134

   1. Initial program 27.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{z - t}{t}\right)}, x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      12. div-sub N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} - \color{blue}{\frac{t}{t}}\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{t}{t}\right)\right)}\right), x\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + -1\right), x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + \color{blue}{\frac{z}{t}}\right), x\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{z}{t}\right)}\right), x\right) \]

      18. /-lowering-/.f64 52.2%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 52.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(\frac{z}{t}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 - \color{blue}{\frac{z}{t}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{z}{t}\right)}\right)\right) \]

      13. /-lowering-/.f64 76.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right)\right) \]

   8. Simplified 76.4%

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

   if -6.5e134 < t < 6.5000000000000002e87

   1. Initial program 84.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \color{blue}{\left(\frac{y - x}{a}\right)}, x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{a}\right), x\right) \]

      5. --lowering--.f64 63.8%

         \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), a\right), x\right) \]

   5. Simplified 63.8%

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

   if 6.5000000000000002e87 < t

   1. Initial program 42.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{t}{a - t}\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{t}{a - t}\right), x\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \color{blue}{\left(a - t\right)}\right), x\right) \]

      16. --lowering--.f64 46.1%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 46.1%

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

   6. Taylor expanded in t around -inf

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

      1. associate-+r+ N/A

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

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

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. distribute-rgt1-in N/A

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

      14. metadata-eval N/A

          \[\leadsto \left(0 \cdot x + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      15. mul0-lft N/A

          \[\leadsto \left(0 + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      16. *-lft-identity N/A

          \[\leadsto \left(0 + 1 \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      17. metadata-eval N/A

          \[\leadsto \left(0 + \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      18. cancel-sign-sub-inv N/A

          \[\leadsto \left(0 - -1 \cdot y\right) + \color{blue}{-1} \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      19. neg-sub0 N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

      22. associate-*r/ N/A

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

   8. Simplified 63.4%

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

Alternative 8: 55.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8e+134)
   (* y (- 1.0 (/ z t)))
   (if (<= t 1.2e+53) (fma y (/ z a) x) (fma a (/ (- y x) t) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8e+134) {
		tmp = y * (1.0 - (z / t));
	} else if (t <= 1.2e+53) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = fma(a, ((y - x) / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8e+134)
		tmp = Float64(y * Float64(1.0 - Float64(z / t)));
	elseif (t <= 1.2e+53)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8e+134], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e+53], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.99999999999999937e134

   1. Initial program 27.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{z - t}{t}\right)}, x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      12. div-sub N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} - \color{blue}{\frac{t}{t}}\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{t}{t}\right)\right)}\right), x\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + -1\right), x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + \color{blue}{\frac{z}{t}}\right), x\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{z}{t}\right)}\right), x\right) \]

      18. /-lowering-/.f64 52.2%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 52.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(\frac{z}{t}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 - \color{blue}{\frac{z}{t}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{z}{t}\right)}\right)\right) \]

      13. /-lowering-/.f64 76.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right)\right) \]

   8. Simplified 76.4%

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

   if -7.99999999999999937e134 < t < 1.2e53

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{a}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(z \cdot \left(y - x\right)\right), \color{blue}{a}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(y - x\right)\right), a\right)\right) \]

      3. --lowering--.f64 62.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, x\right)\right), a\right)\right) \]

   5. Simplified 62.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{y}\right), a\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 58.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{\left(\frac{z}{a}\right)}, x\right) \]

      5. /-lowering-/.f64 61.1%

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{a}\right), x\right) \]

   10. Applied egg-rr 61.1%

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

   if 1.2e53 < t

   1. Initial program 45.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{t}{a - t}\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{t}{a - t}\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{t}{a - t}\right), x\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{t}}{a - t}\right), x\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \color{blue}{\left(a - t\right)}\right), x\right) \]

      16. --lowering--.f64 44.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(t, \mathsf{\_.f64}\left(a, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 44.8%

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

   6. Taylor expanded in t around -inf

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

      1. associate-+r+ N/A

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

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

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. distribute-rgt1-in N/A

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

      14. metadata-eval N/A

          \[\leadsto \left(0 \cdot x + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      15. mul0-lft N/A

          \[\leadsto \left(0 + y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      16. *-lft-identity N/A

          \[\leadsto \left(0 + 1 \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      17. metadata-eval N/A

          \[\leadsto \left(0 + \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) + -1 \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      18. cancel-sign-sub-inv N/A

          \[\leadsto \left(0 - -1 \cdot y\right) + \color{blue}{-1} \cdot \frac{a \cdot \left(x - y\right)}{t} \]

      19. neg-sub0 N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

      22. associate-*r/ N/A

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

   8. Simplified 57.3%

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

Alternative 9: 55.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- 1.0 (/ z t)))))
   (if (<= t -6.5e+134) t_1 (if (<= t 2.1e+52) (fma y (/ z a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (1.0 - (z / t));
	double tmp;
	if (t <= -6.5e+134) {
		tmp = t_1;
	} else if (t <= 2.1e+52) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(1.0 - Float64(z / t)))
	tmp = 0.0
	if (t <= -6.5e+134)
		tmp = t_1;
	elseif (t <= 2.1e+52)
		tmp = fma(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[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.5e+134], t$95$1, If[LessEqual[t, 2.1e+52], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5e134 or 2.1e52 < t

   1. Initial program 38.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{\left(\frac{z - t}{t}\right)}, x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - y\right), \left(\frac{\color{blue}{z} - t}{t}\right), x\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\color{blue}{z - t}}{t}\right), x\right) \]

      12. div-sub N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} - \color{blue}{\frac{t}{t}}\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{t}{t}\right)\right)}\right), x\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{z}{t} + -1\right), x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + \color{blue}{\frac{z}{t}}\right), x\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{z}{t}\right)}\right), x\right) \]

      18. /-lowering-/.f64 49.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right), x\right) \]

   5. Simplified 49.8%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(\frac{z}{t}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 - \color{blue}{\frac{z}{t}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{z}{t}\right)}\right)\right) \]

      13. /-lowering-/.f64 63.1%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right)\right) \]

   8. Simplified 63.1%

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

   if -6.5e134 < t < 2.1e52

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{a}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(z \cdot \left(y - x\right)\right), \color{blue}{a}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(y - x\right)\right), a\right)\right) \]

      3. --lowering--.f64 62.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, x\right)\right), a\right)\right) \]

   5. Simplified 62.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{y}\right), a\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 58.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{\left(\frac{z}{a}\right)}, x\right) \]

      5. /-lowering-/.f64 61.1%

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{a}\right), x\right) \]

   10. Applied egg-rr 61.1%

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

Alternative 10: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+134}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8.5e+134) y (if (<= t 4.7e+54) (fma y (/ z a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8.5e+134) {
		tmp = y;
	} else if (t <= 4.7e+54) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8.5e+134)
		tmp = y;
	elseif (t <= 4.7e+54)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8.5e+134], y, If[LessEqual[t, 4.7e+54], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -8.50000000000000024e134 or 4.69999999999999993e54 < t

   1. Initial program 38.4%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Simplified 58.8%

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

   if -8.50000000000000024e134 < t < 4.69999999999999993e54

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{a}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(z \cdot \left(y - x\right)\right), \color{blue}{a}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(y - x\right)\right), a\right)\right) \]

      3. --lowering--.f64 62.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, x\right)\right), a\right)\right) \]

   5. Simplified 62.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{y}\right), a\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 58.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{\left(\frac{z}{a}\right)}, x\right) \]

      5. /-lowering-/.f64 61.1%

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{a}\right), x\right) \]

   10. Applied egg-rr 61.1%

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

Alternative 11: 37.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+129}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.3e+129) y (if (<= t 1.15e+54) x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.3e+129) {
		tmp = y;
	} else if (t <= 1.15e+54) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-1.3d+129)) then
        tmp = y
    else if (t <= 1.15d+54) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.3e+129) {
		tmp = y;
	} else if (t <= 1.15e+54) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.3e+129:
		tmp = y
	elif t <= 1.15e+54:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.3e+129)
		tmp = y;
	elseif (t <= 1.15e+54)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.3e+129)
		tmp = y;
	elseif (t <= 1.15e+54)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.3e+129], y, If[LessEqual[t, 1.15e+54], x, y]]

Derivation

1. Split input into 2 regimes

2. if t < -1.30000000000000006e129 or 1.14999999999999997e54 < t

   1. Initial program 38.1%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Simplified 58.3%

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

   if -1.30000000000000006e129 < t < 1.14999999999999997e54

   1. Initial program 86.5%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 33.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 24.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 23.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 86.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024193 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))