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

Percentage Accurate: 67.5% → 89.5%

Time: 10.4s

Alternatives: 18

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1}{t - a} \cdot \left(z - t\right), y - x, x\right)\\ t_2 := \frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;y - \frac{\mathsf{fma}\left(\left(z - a\right) \cdot \frac{y - x}{t}, a, \left(z - a\right) \cdot \left(y - x\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (* (/ -1.0 (- t a)) (- z t)) (- y x) x))
        (t_2 (+ (/ (* (- z t) (- y x)) (- a t)) x)))
   (if (<= t_2 -1e-223)
     t_1
     (if (<= t_2 0.0)
       (- y (/ (fma (* (- z a) (/ (- y x) t)) a (* (- z a) (- y x))) t))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((-1.0 / (t - a)) * (z - t)), (y - x), x);
	double t_2 = (((z - t) * (y - x)) / (a - t)) + x;
	double tmp;
	if (t_2 <= -1e-223) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = y - (fma(((z - a) * ((y - x) / t)), a, ((z - a) * (y - x))) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-1.0 / Float64(t - a)) * Float64(z - t)), Float64(y - x), x)
	t_2 = Float64(Float64(Float64(Float64(z - t) * Float64(y - x)) / Float64(a - t)) + x)
	tmp = 0.0
	if (t_2 <= -1e-223)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(y - Float64(fma(Float64(Float64(z - a) * Float64(Float64(y - x) / t)), a, Float64(Float64(z - a) * Float64(y - x))) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(-1.0 / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z - t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-223], t$95$1, If[LessEqual[t$95$2, 0.0], N[(y - N[(N[(N[(N[(z - a), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * a + N[(N[(z - a), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -9.9999999999999997e-224 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}} \cdot \left(z - t\right), y - x, x\right) \]

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 90.5

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

   4. Applied rewrites 90.5%

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

   if -9.9999999999999997e-224 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 99.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x \leq -1 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t - a} \cdot \left(z - t\right), y - x, x\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x \leq 0:\\ \;\;\;\;y - \frac{\mathsf{fma}\left(\left(z - a\right) \cdot \frac{y - x}{t}, a, \left(z - a\right) \cdot \left(y - x\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t - a} \cdot \left(z - t\right), y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (/ (* (- z t) (- y x)) (- a t)) x)))
   (if (<= t_1 (- INFINITY))
     (fma (/ (- x y) t) (- z a) y)
     (if (<= t_1 -1e-223)
       t_1
       (if (<= t_1 0.0)
         (fma (/ x t) (- z a) y)
         (if (<= t_1 5e+280) t_1 (* (/ (- z t) (- a t)) y)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (((z - t) * (y - x)) / (a - t)) + x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((x - y) / t), (z - a), y);
	} else if (t_1 <= -1e-223) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma((x / t), (z - a), y);
	} else if (t_1 <= 5e+280) {
		tmp = t_1;
	} else {
		tmp = ((z - t) / (a - t)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(Float64(z - t) * Float64(y - x)) / Float64(a - t)) + x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(x - y) / t), Float64(z - a), y);
	elseif (t_1 <= -1e-223)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(x / t), Float64(z - a), y);
	elseif (t_1 <= 5e+280)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(z - t) / Float64(a - t)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[(z - t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t$95$1, -1e-223], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t$95$1, 5e+280], t$95$1, N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -inf.0

   1. Initial program 30.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 74.5%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -9.9999999999999997e-224 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 5.0000000000000002e280

   1. Initial program 99.6%

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

   if -9.9999999999999997e-224 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.9%

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

   if 5.0000000000000002e280 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 33.9%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 68.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x \leq -1 \cdot 10^{-223}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1}{t - a} \cdot \left(z - t\right), y - x, x\right)\\ t_2 := \frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (* (/ -1.0 (- t a)) (- z t)) (- y x) x))
        (t_2 (+ (/ (* (- z t) (- y x)) (- a t)) x)))
   (if (<= t_2 -1e-223) t_1 (if (<= t_2 0.0) (fma (/ x t) (- z a) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((-1.0 / (t - a)) * (z - t)), (y - x), x);
	double t_2 = (((z - t) * (y - x)) / (a - t)) + x;
	double tmp;
	if (t_2 <= -1e-223) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma((x / t), (z - a), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-1.0 / Float64(t - a)) * Float64(z - t)), Float64(y - x), x)
	t_2 = Float64(Float64(Float64(Float64(z - t) * Float64(y - x)) / Float64(a - t)) + x)
	tmp = 0.0
	if (t_2 <= -1e-223)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(x / t), Float64(z - a), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(-1.0 / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z - t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-223], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -9.9999999999999997e-224 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}} \cdot \left(z - t\right), y - x, x\right) \]

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 90.5

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

   4. Applied rewrites 90.5%

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

   if -9.9999999999999997e-224 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x \leq -1 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t - a} \cdot \left(z - t\right), y - x, x\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot \left(y - x\right)}{a - t} + x \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t - a} \cdot \left(z - t\right), y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t} \cdot y\\ t_2 := \mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+100}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.096:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) (- a t)) y)) (t_2 (fma (/ x t) (- z a) y)))
   (if (<= t -4.5e+100)
     t_2
     (if (<= t -1.45e-17)
       t_1
       (if (<= t 0.096) (fma (/ (- y x) a) z x) (if (<= t 6e+154) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / (a - t)) * y;
	double t_2 = fma((x / t), (z - a), y);
	double tmp;
	if (t <= -4.5e+100) {
		tmp = t_2;
	} else if (t <= -1.45e-17) {
		tmp = t_1;
	} else if (t <= 0.096) {
		tmp = fma(((y - x) / a), z, x);
	} else if (t <= 6e+154) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) / Float64(a - t)) * y)
	t_2 = fma(Float64(x / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -4.5e+100)
		tmp = t_2;
	elseif (t <= -1.45e-17)
		tmp = t_1;
	elseif (t <= 0.096)
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	elseif (t <= 6e+154)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -4.5e+100], t$95$2, If[LessEqual[t, -1.45e-17], t$95$1, If[LessEqual[t, 0.096], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t, 6e+154], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.50000000000000036e100 or 6.00000000000000052e154 < t

   1. Initial program 34.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 75.9%

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

   if -4.50000000000000036e100 < t < -1.4500000000000001e-17 or 0.096000000000000002 < t < 6.00000000000000052e154

   1. Initial program 65.2%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 70.2%

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

   if -1.4500000000000001e-17 < t < 0.096000000000000002

   1. Initial program 92.7%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 78.4

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

   5. Applied rewrites 78.4%

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

Alternative 5: 69.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - t} \cdot \left(z - t\right)\\ t_2 := \mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+100}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.096:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- a t)) (- z t))) (t_2 (fma (/ x t) (- z a) y)))
   (if (<= t -4.5e+100)
     t_2
     (if (<= t -1.45e-17)
       t_1
       (if (<= t 0.096) (fma (/ (- y x) a) z x) (if (<= t 6e+154) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (a - t)) * (z - t);
	double t_2 = fma((x / t), (z - a), y);
	double tmp;
	if (t <= -4.5e+100) {
		tmp = t_2;
	} else if (t <= -1.45e-17) {
		tmp = t_1;
	} else if (t <= 0.096) {
		tmp = fma(((y - x) / a), z, x);
	} else if (t <= 6e+154) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(a - t)) * Float64(z - t))
	t_2 = fma(Float64(x / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -4.5e+100)
		tmp = t_2;
	elseif (t <= -1.45e-17)
		tmp = t_1;
	elseif (t <= 0.096)
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	elseif (t <= 6e+154)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -4.5e+100], t$95$2, If[LessEqual[t, -1.45e-17], t$95$1, If[LessEqual[t, 0.096], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t, 6e+154], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.50000000000000036e100 or 6.00000000000000052e154 < t

   1. Initial program 34.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 75.9%

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

   if -4.50000000000000036e100 < t < -1.4500000000000001e-17 or 0.096000000000000002 < t < 6.00000000000000052e154

   1. Initial program 65.2%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if -1.4500000000000001e-17 < t < 0.096000000000000002

   1. Initial program 92.7%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 78.4

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

   5. Applied rewrites 78.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \mathbf{elif}\;t \leq 0.096:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(x - y\right) \cdot z}{t}\\ t_2 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-134}:\\ \;\;\;\;\frac{z}{a} \cdot \left(y - x\right)\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- x y) z) t)) (t_2 (+ (- y x) x)))
   (if (<= t -1.02e+81)
     t_2
     (if (<= t -1.7e-35)
       t_1
       (if (<= t 1.15e-134)
         (* (/ z a) (- y x))
         (if (<= t 2.25e+79) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x - y) * z) / t;
	double t_2 = (y - x) + x;
	double tmp;
	if (t <= -1.02e+81) {
		tmp = t_2;
	} else if (t <= -1.7e-35) {
		tmp = t_1;
	} else if (t <= 1.15e-134) {
		tmp = (z / a) * (y - x);
	} else if (t <= 2.25e+79) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = ((x - y) * z) / t
    t_2 = (y - x) + x
    if (t <= (-1.02d+81)) then
        tmp = t_2
    else if (t <= (-1.7d-35)) then
        tmp = t_1
    else if (t <= 1.15d-134) then
        tmp = (z / a) * (y - x)
    else if (t <= 2.25d+79) then
        tmp = t_1
    else
        tmp = t_2
    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) * z) / t;
	double t_2 = (y - x) + x;
	double tmp;
	if (t <= -1.02e+81) {
		tmp = t_2;
	} else if (t <= -1.7e-35) {
		tmp = t_1;
	} else if (t <= 1.15e-134) {
		tmp = (z / a) * (y - x);
	} else if (t <= 2.25e+79) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((x - y) * z) / t
	t_2 = (y - x) + x
	tmp = 0
	if t <= -1.02e+81:
		tmp = t_2
	elif t <= -1.7e-35:
		tmp = t_1
	elif t <= 1.15e-134:
		tmp = (z / a) * (y - x)
	elif t <= 2.25e+79:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x - y) * z) / t)
	t_2 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -1.02e+81)
		tmp = t_2;
	elseif (t <= -1.7e-35)
		tmp = t_1;
	elseif (t <= 1.15e-134)
		tmp = Float64(Float64(z / a) * Float64(y - x));
	elseif (t <= 2.25e+79)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((x - y) * z) / t;
	t_2 = (y - x) + x;
	tmp = 0.0;
	if (t <= -1.02e+81)
		tmp = t_2;
	elseif (t <= -1.7e-35)
		tmp = t_1;
	elseif (t <= 1.15e-134)
		tmp = (z / a) * (y - x);
	elseif (t <= 2.25e+79)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.02e+81], t$95$2, If[LessEqual[t, -1.7e-35], t$95$1, If[LessEqual[t, 1.15e-134], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.25e+79], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.01999999999999992e81 or 2.24999999999999997e79 < t

   1. Initial program 36.7%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 40.0

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

   5. Applied rewrites 40.0%

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

   if -1.01999999999999992e81 < t < -1.7000000000000001e-35 or 1.15e-134 < t < 2.24999999999999997e79

   1. Initial program 84.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 37.5%

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

   if -1.7000000000000001e-35 < t < 1.15e-134

   1. Initial program 93.4%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 51.1%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+81}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot z}{t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-134}:\\ \;\;\;\;\frac{z}{a} \cdot \left(y - x\right)\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(x - y\right) \cdot z}{t}\\ t_2 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- x y) z) t)) (t_2 (+ (- y x) x)))
   (if (<= t -1.02e+81)
     t_2
     (if (<= t -1.35e-35)
       t_1
       (if (<= t 9.5e-135)
         (/ (* z (- y x)) a)
         (if (<= t 2.25e+79) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x - y) * z) / t;
	double t_2 = (y - x) + x;
	double tmp;
	if (t <= -1.02e+81) {
		tmp = t_2;
	} else if (t <= -1.35e-35) {
		tmp = t_1;
	} else if (t <= 9.5e-135) {
		tmp = (z * (y - x)) / a;
	} else if (t <= 2.25e+79) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = ((x - y) * z) / t
    t_2 = (y - x) + x
    if (t <= (-1.02d+81)) then
        tmp = t_2
    else if (t <= (-1.35d-35)) then
        tmp = t_1
    else if (t <= 9.5d-135) then
        tmp = (z * (y - x)) / a
    else if (t <= 2.25d+79) then
        tmp = t_1
    else
        tmp = t_2
    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) * z) / t;
	double t_2 = (y - x) + x;
	double tmp;
	if (t <= -1.02e+81) {
		tmp = t_2;
	} else if (t <= -1.35e-35) {
		tmp = t_1;
	} else if (t <= 9.5e-135) {
		tmp = (z * (y - x)) / a;
	} else if (t <= 2.25e+79) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((x - y) * z) / t
	t_2 = (y - x) + x
	tmp = 0
	if t <= -1.02e+81:
		tmp = t_2
	elif t <= -1.35e-35:
		tmp = t_1
	elif t <= 9.5e-135:
		tmp = (z * (y - x)) / a
	elif t <= 2.25e+79:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x - y) * z) / t)
	t_2 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -1.02e+81)
		tmp = t_2;
	elseif (t <= -1.35e-35)
		tmp = t_1;
	elseif (t <= 9.5e-135)
		tmp = Float64(Float64(z * Float64(y - x)) / a);
	elseif (t <= 2.25e+79)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((x - y) * z) / t;
	t_2 = (y - x) + x;
	tmp = 0.0;
	if (t <= -1.02e+81)
		tmp = t_2;
	elseif (t <= -1.35e-35)
		tmp = t_1;
	elseif (t <= 9.5e-135)
		tmp = (z * (y - x)) / a;
	elseif (t <= 2.25e+79)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.02e+81], t$95$2, If[LessEqual[t, -1.35e-35], t$95$1, If[LessEqual[t, 9.5e-135], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 2.25e+79], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.01999999999999992e81 or 2.24999999999999997e79 < t

   1. Initial program 36.7%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 40.0

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

   5. Applied rewrites 40.0%

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

   if -1.01999999999999992e81 < t < -1.3499999999999999e-35 or 9.50000000000000007e-135 < t < 2.24999999999999997e79

   1. Initial program 84.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 37.5%

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

   if -1.3499999999999999e-35 < t < 9.50000000000000007e-135

   1. Initial program 93.4%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 48.1%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+81}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot z}{t}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -1.28 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{y}{a - t} \cdot z\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- y x) x)))
   (if (<= t -1.28e+29)
     t_1
     (if (<= t -3.3e-162)
       (* (/ y (- a t)) z)
       (if (<= t 7.4e-153)
         (/ (* z (- y x)) a)
         (if (<= t 2e+75) (* (/ (- z t) a) y) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -1.28e+29) {
		tmp = t_1;
	} else if (t <= -3.3e-162) {
		tmp = (y / (a - t)) * z;
	} else if (t <= 7.4e-153) {
		tmp = (z * (y - x)) / a;
	} else if (t <= 2e+75) {
		tmp = ((z - t) / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - x) + x
    if (t <= (-1.28d+29)) then
        tmp = t_1
    else if (t <= (-3.3d-162)) then
        tmp = (y / (a - t)) * z
    else if (t <= 7.4d-153) then
        tmp = (z * (y - x)) / a
    else if (t <= 2d+75) then
        tmp = ((z - t) / a) * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -1.28e+29) {
		tmp = t_1;
	} else if (t <= -3.3e-162) {
		tmp = (y / (a - t)) * z;
	} else if (t <= 7.4e-153) {
		tmp = (z * (y - x)) / a;
	} else if (t <= 2e+75) {
		tmp = ((z - t) / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - x) + x
	tmp = 0
	if t <= -1.28e+29:
		tmp = t_1
	elif t <= -3.3e-162:
		tmp = (y / (a - t)) * z
	elif t <= 7.4e-153:
		tmp = (z * (y - x)) / a
	elif t <= 2e+75:
		tmp = ((z - t) / a) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -1.28e+29)
		tmp = t_1;
	elseif (t <= -3.3e-162)
		tmp = Float64(Float64(y / Float64(a - t)) * z);
	elseif (t <= 7.4e-153)
		tmp = Float64(Float64(z * Float64(y - x)) / a);
	elseif (t <= 2e+75)
		tmp = Float64(Float64(Float64(z - t) / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - x) + x;
	tmp = 0.0;
	if (t <= -1.28e+29)
		tmp = t_1;
	elseif (t <= -3.3e-162)
		tmp = (y / (a - t)) * z;
	elseif (t <= 7.4e-153)
		tmp = (z * (y - x)) / a;
	elseif (t <= 2e+75)
		tmp = ((z - t) / a) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.28e+29], t$95$1, If[LessEqual[t, -3.3e-162], N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t, 7.4e-153], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 2e+75], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.28e29 or 1.99999999999999985e75 < t

   1. Initial program 40.3%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 37.8

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

   5. Applied rewrites 37.8%

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

   if -1.28e29 < t < -3.30000000000000013e-162

   1. Initial program 92.5%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 34.7%

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

   10. Applied rewrites 36.9%

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

   if -3.30000000000000013e-162 < t < 7.4000000000000005e-153

   1. Initial program 96.7%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 54.0%

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

   if 7.4000000000000005e-153 < t < 1.99999999999999985e75

   1. Initial program 81.7%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 40.5%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.28 \cdot 10^{+29}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{y}{a - t} \cdot z\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{t}, -a, y\right)\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-134}:\\ \;\;\;\;\frac{z}{a} \cdot \left(y - x\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x t) (- a) y)))
   (if (<= t -6.4e-18)
     t_1
     (if (<= t 1.15e-134)
       (* (/ z a) (- y x))
       (if (<= t 1.8e+70) (/ (* (- x y) z) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x / t), -a, y);
	double tmp;
	if (t <= -6.4e-18) {
		tmp = t_1;
	} else if (t <= 1.15e-134) {
		tmp = (z / a) * (y - x);
	} else if (t <= 1.8e+70) {
		tmp = ((x - y) * z) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x / t), Float64(-a), y)
	tmp = 0.0
	if (t <= -6.4e-18)
		tmp = t_1;
	elseif (t <= 1.15e-134)
		tmp = Float64(Float64(z / a) * Float64(y - x));
	elseif (t <= 1.8e+70)
		tmp = Float64(Float64(Float64(x - y) * z) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / t), $MachinePrecision] * (-a) + y), $MachinePrecision]}, If[LessEqual[t, -6.4e-18], t$95$1, If[LessEqual[t, 1.15e-134], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.8e+70], N[(N[(N[(x - y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.3999999999999998e-18 or 1.8e70 < t

   1. Initial program 45.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 51.9%

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

   if -6.3999999999999998e-18 < t < 1.15e-134

   1. Initial program 93.7%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 53.7

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 48.6%

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

   if 1.15e-134 < t < 1.8e70

   1. Initial program 81.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 41.3%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, -a, y\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-134}:\\ \;\;\;\;\frac{z}{a} \cdot \left(y - x\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, -a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -2 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- y x) x)))
   (if (<= t -2e-17)
     t_1
     (if (<= t 7.4e-153)
       (/ (* z (- y x)) a)
       (if (<= t 2e+75) (* (/ (- z t) a) y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -2e-17) {
		tmp = t_1;
	} else if (t <= 7.4e-153) {
		tmp = (z * (y - x)) / a;
	} else if (t <= 2e+75) {
		tmp = ((z - t) / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - x) + x
    if (t <= (-2d-17)) then
        tmp = t_1
    else if (t <= 7.4d-153) then
        tmp = (z * (y - x)) / a
    else if (t <= 2d+75) then
        tmp = ((z - t) / a) * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -2e-17) {
		tmp = t_1;
	} else if (t <= 7.4e-153) {
		tmp = (z * (y - x)) / a;
	} else if (t <= 2e+75) {
		tmp = ((z - t) / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - x) + x
	tmp = 0
	if t <= -2e-17:
		tmp = t_1
	elif t <= 7.4e-153:
		tmp = (z * (y - x)) / a
	elif t <= 2e+75:
		tmp = ((z - t) / a) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -2e-17)
		tmp = t_1;
	elseif (t <= 7.4e-153)
		tmp = Float64(Float64(z * Float64(y - x)) / a);
	elseif (t <= 2e+75)
		tmp = Float64(Float64(Float64(z - t) / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - x) + x;
	tmp = 0.0;
	if (t <= -2e-17)
		tmp = t_1;
	elseif (t <= 7.4e-153)
		tmp = (z * (y - x)) / a;
	elseif (t <= 2e+75)
		tmp = ((z - t) / a) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -2e-17], t$95$1, If[LessEqual[t, 7.4e-153], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 2e+75], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.00000000000000014e-17 or 1.99999999999999985e75 < t

   1. Initial program 45.4%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 35.7

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

   5. Applied rewrites 35.7%

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

   if -2.00000000000000014e-17 < t < 7.4000000000000005e-153

   1. Initial program 94.4%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 46.1%

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

   if 7.4000000000000005e-153 < t < 1.99999999999999985e75

   1. Initial program 81.7%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 40.5%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 21000:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) (- z a) y)))
   (if (<= t -2.8e+33)
     t_1
     (if (<= t 21000.0) (+ (/ (* z (- y x)) (- a t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), (z - a), y);
	double tmp;
	if (t <= -2.8e+33) {
		tmp = t_1;
	} else if (t <= 21000.0) {
		tmp = ((z * (y - x)) / (a - t)) + 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 <= -2.8e+33)
		tmp = t_1;
	elseif (t <= 21000.0)
		tmp = Float64(Float64(Float64(z * Float64(y - x)) / Float64(a - t)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.8e+33], t$95$1, If[LessEqual[t, 21000.0], N[(N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8000000000000001e33 or 21000 < t

   1. Initial program 40.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 75.9%

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

   if -2.8000000000000001e33 < t < 21000

   1. Initial program 93.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{elif}\;t \leq 21000:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) (- z a) y)))
   (if (<= t -2.55e+33) t_1 (if (<= t 3.4e-28) (fma (/ (- y x) a) z 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 <= -2.55e+33) {
		tmp = t_1;
	} else if (t <= 3.4e-28) {
		tmp = fma(((y - x) / a), z, 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 <= -2.55e+33)
		tmp = t_1;
	elseif (t <= 3.4e-28)
		tmp = fma(Float64(Float64(y - x) / a), z, 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, -2.55e+33], t$95$1, If[LessEqual[t, 3.4e-28], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.5499999999999999e33 or 3.4000000000000001e-28 < t

   1. Initial program 41.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 74.8%

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

   if -2.5499999999999999e33 < t < 3.4000000000000001e-28

   1. Initial program 94.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

Alternative 13: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x t) (- z a) y)))
   (if (<= t -2.8e+33) t_1 (if (<= t 3.5e-27) (fma (/ (- y x) a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x / t), (z - a), y);
	double tmp;
	if (t <= -2.8e+33) {
		tmp = t_1;
	} else if (t <= 3.5e-27) {
		tmp = fma(((y - x) / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -2.8e+33)
		tmp = t_1;
	elseif (t <= 3.5e-27)
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.8e+33], t$95$1, If[LessEqual[t, 3.5e-27], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8000000000000001e33 or 3.5000000000000001e-27 < t

   1. Initial program 41.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.1%

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

   if -2.8000000000000001e33 < t < 3.5000000000000001e-27

   1. Initial program 94.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

Alternative 14: 56.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{z}{a} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) z y)))
   (if (<= t -1.3e-35) t_1 (if (<= t 1.2e-134) (* (/ z a) (- y x)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), z, y);
	double tmp;
	if (t <= -1.3e-35) {
		tmp = t_1;
	} else if (t <= 1.2e-134) {
		tmp = (z / a) * (y - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - y) / t), z, y)
	tmp = 0.0
	if (t <= -1.3e-35)
		tmp = t_1;
	elseif (t <= 1.2e-134)
		tmp = Float64(Float64(z / a) * Float64(y - 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] * z + y), $MachinePrecision]}, If[LessEqual[t, -1.3e-35], t$95$1, If[LessEqual[t, 1.2e-134], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.30000000000000002e-35 or 1.20000000000000005e-134 < t

   1. Initial program 54.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 61.9%

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

   if -1.30000000000000002e-35 < t < 1.20000000000000005e-134

   1. Initial program 93.4%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 51.1%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{z}{a} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -2 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+89}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- y x) x)))
   (if (<= t -2e-17) t_1 (if (<= t 2.25e+89) (/ (* z (- y x)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -2e-17) {
		tmp = t_1;
	} else if (t <= 2.25e+89) {
		tmp = (z * (y - x)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - x) + x
    if (t <= (-2d-17)) then
        tmp = t_1
    else if (t <= 2.25d+89) then
        tmp = (z * (y - x)) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -2e-17) {
		tmp = t_1;
	} else if (t <= 2.25e+89) {
		tmp = (z * (y - x)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - x) + x
	tmp = 0
	if t <= -2e-17:
		tmp = t_1
	elif t <= 2.25e+89:
		tmp = (z * (y - x)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -2e-17)
		tmp = t_1;
	elseif (t <= 2.25e+89)
		tmp = Float64(Float64(z * Float64(y - x)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - x) + x;
	tmp = 0.0;
	if (t <= -2e-17)
		tmp = t_1;
	elseif (t <= 2.25e+89)
		tmp = (z * (y - x)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -2e-17], t$95$1, If[LessEqual[t, 2.25e+89], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.00000000000000014e-17 or 2.25e89 < t

   1. Initial program 45.8%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 36.0

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

   5. Applied rewrites 36.0%

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

   if -2.00000000000000014e-17 < t < 2.25e89

   1. Initial program 89.6%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 39.8%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+89}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- y x) x)))
   (if (<= t -1.8e-19) t_1 (if (<= t 1.95e+89) (* (/ z a) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -1.8e-19) {
		tmp = t_1;
	} else if (t <= 1.95e+89) {
		tmp = (z / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - x) + x
    if (t <= (-1.8d-19)) then
        tmp = t_1
    else if (t <= 1.95d+89) then
        tmp = (z / a) * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -1.8e-19) {
		tmp = t_1;
	} else if (t <= 1.95e+89) {
		tmp = (z / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - x) + x
	tmp = 0
	if t <= -1.8e-19:
		tmp = t_1
	elif t <= 1.95e+89:
		tmp = (z / a) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -1.8e-19)
		tmp = t_1;
	elseif (t <= 1.95e+89)
		tmp = Float64(Float64(z / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - x) + x;
	tmp = 0.0;
	if (t <= -1.8e-19)
		tmp = t_1;
	elseif (t <= 1.95e+89)
		tmp = (z / a) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.8e-19], t$95$1, If[LessEqual[t, 1.95e+89], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8000000000000001e-19 or 1.95000000000000005e89 < t

   1. Initial program 46.2%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 35.7

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

   5. Applied rewrites 35.7%

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

   if -1.8000000000000001e-19 < t < 1.95000000000000005e89

   1. Initial program 89.6%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 32.6%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-19}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 20.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

3. Taylor expanded in t around inf

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

   1. lower--.f64 20.3

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

5. Applied rewrites 20.3%

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

6. Final simplification 20.3%

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

Alternative 18: 2.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

3. Taylor expanded in t around inf

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

   1. lower--.f64 20.3

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

5. Applied rewrites 20.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 2.8%

   \[\leadsto x + \left(-x\right) \]

9. Final simplification 2.8%

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

Developer Target 1: 86.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024240 
(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))))