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

Percentage Accurate: 67.6% → 85.3%

Time: 13.3s

Alternatives: 15

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: 67.6% 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: 85.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ t_2 := \frac{y}{x} - 1\\ t_3 := \mathsf{fma}\left(\frac{z - t}{a - t}, t\_2, 1\right) \cdot x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-267}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-245}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t\_2 \cdot \left(z - a\right)}{-t}, y\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t))))
        (t_2 (- (/ y x) 1.0))
        (t_3 (* (fma (/ (- z t) (- a t)) t_2 1.0) x)))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 -2e-267)
       t_1
       (if (<= t_1 5e-245)
         (fma x (/ (* t_2 (- z a)) (- t)) y)
         (if (<= t_1 2e+282) t_1 t_3))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double t_2 = (y / x) - 1.0;
	double t_3 = fma(((z - t) / (a - t)), t_2, 1.0) * x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= -2e-267) {
		tmp = t_1;
	} else if (t_1 <= 5e-245) {
		tmp = fma(x, ((t_2 * (z - a)) / -t), y);
	} else if (t_1 <= 2e+282) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	t_2 = Float64(Float64(y / x) - 1.0)
	t_3 = Float64(fma(Float64(Float64(z - t) / Float64(a - t)), t_2, 1.0) * x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= -2e-267)
		tmp = t_1;
	elseif (t_1 <= 5e-245)
		tmp = fma(x, Float64(Float64(t_2 * Float64(z - a)) / Float64(-t)), y);
	elseif (t_1 <= 2e+282)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / x), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, -2e-267], t$95$1, If[LessEqual[t$95$1, 5e-245], N[(x * N[(N[(t$95$2 * N[(z - a), $MachinePrecision]), $MachinePrecision] / (-t)), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t$95$1, 2e+282], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -inf.0 or 2.00000000000000007e282 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.2%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -2e-267 or 4.9999999999999997e-245 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 2.00000000000000007e282

   1. Initial program 94.4%

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

   if -2e-267 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 4.9999999999999997e-245

   1. Initial program 14.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 14.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, \frac{y}{x} - 1, 1\right) \cdot x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2 \cdot 10^{-267}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 5 \cdot 10^{-245}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\left(\frac{y}{x} - 1\right) \cdot \left(z - a\right)}{-t}, y\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 2 \cdot 10^{+282}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, \frac{y}{x} - 1, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-y\right) + x\right) \cdot \frac{z - a}{t}\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{+122} \lor \neg \left(t \leq 8.8 \cdot 10^{+69}\right):\\ \;\;\;\;y + \mathsf{fma}\left(t\_1, \frac{a}{t}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (+ (- y) x) (/ (- z a) t))))
   (if (or (<= t -1.85e+122) (not (<= t 8.8e+69)))
     (+ y (fma t_1 (/ a t) t_1))
     (+ x (/ (* (- y x) (- z t)) (- a t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-y + x) * ((z - a) / t);
	double tmp;
	if ((t <= -1.85e+122) || !(t <= 8.8e+69)) {
		tmp = y + fma(t_1, (a / t), t_1);
	} else {
		tmp = x + (((y - x) * (z - t)) / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-y) + x) * Float64(Float64(z - a) / t))
	tmp = 0.0
	if ((t <= -1.85e+122) || !(t <= 8.8e+69))
		tmp = Float64(y + fma(t_1, Float64(a / t), t_1));
	else
		tmp = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-y) + x), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -1.85e+122], N[Not[LessEqual[t, 8.8e+69]], $MachinePrecision]], N[(y + N[(t$95$1 * N[(a / t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8499999999999998e122 or 8.8000000000000006e69 < t

   1. Initial program 34.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 12.3

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

   8. Applied rewrites 12.3%

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

   9. Taylor expanded in t around inf

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

       1. fp-cancel-sub-sign-inv N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. associate-+l- N/A

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

       8. div-sub N/A

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

       9. lower--.f64 N/A

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

       10. div-sub N/A

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

       11. associate-/l* N/A

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

       12. associate-/l* N/A

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

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

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 83.2

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

   11. Applied rewrites 83.2%

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

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} + \frac{a \cdot \left(-1 \cdot \left(z \cdot \left(y - x\right)\right) - -1 \cdot \left(a \cdot \left(y - x\right)\right)\right)}{{t}^{2}}\right)\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{y + \left(\left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} + \frac{a \cdot \left(-1 \cdot \left(z \cdot \left(y - x\right)\right) - -1 \cdot \left(a \cdot \left(y - x\right)\right)\right)}{{t}^{2}}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

       2. fp-cancel-sub-sign-inv N/A

          \[\leadsto y + \color{blue}{\left(\left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} + \frac{a \cdot \left(-1 \cdot \left(z \cdot \left(y - x\right)\right) - -1 \cdot \left(a \cdot \left(y - x\right)\right)\right)}{{t}^{2}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

       3. metadata-eval N/A

          \[\leadsto y + \left(\left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} + \frac{a \cdot \left(-1 \cdot \left(z \cdot \left(y - x\right)\right) - -1 \cdot \left(a \cdot \left(y - x\right)\right)\right)}{{t}^{2}}\right) + \color{blue}{1} \cdot \frac{a \cdot \left(y - x\right)}{t}\right) \]

       4. *-lft-identity N/A

          \[\leadsto y + \left(\left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} + \frac{a \cdot \left(-1 \cdot \left(z \cdot \left(y - x\right)\right) - -1 \cdot \left(a \cdot \left(y - x\right)\right)\right)}{{t}^{2}}\right) + \color{blue}{\frac{a \cdot \left(y - x\right)}{t}}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \color{blue}{y + \left(\left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} + \frac{a \cdot \left(-1 \cdot \left(z \cdot \left(y - x\right)\right) - -1 \cdot \left(a \cdot \left(y - x\right)\right)\right)}{{t}^{2}}\right) + \frac{a \cdot \left(y - x\right)}{t}\right)} \]

       6. *-lft-identity N/A

          \[\leadsto y + \left(\left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} + \frac{a \cdot \left(-1 \cdot \left(z \cdot \left(y - x\right)\right) - -1 \cdot \left(a \cdot \left(y - x\right)\right)\right)}{{t}^{2}}\right) + \color{blue}{1 \cdot \frac{a \cdot \left(y - x\right)}{t}}\right) \]

   14. Applied rewrites 85.6%

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

   if -1.8499999999999998e122 < t < 8.8000000000000006e69

   1. Initial program 86.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+122} \lor \neg \left(t \leq 8.8 \cdot 10^{+69}\right):\\ \;\;\;\;y + \mathsf{fma}\left(\left(\left(-y\right) + x\right) \cdot \frac{z - a}{t}, \frac{a}{t}, \left(\left(-y\right) + x\right) \cdot \frac{z - a}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+122} \lor \neg \left(t \leq 10^{+66}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.85e+122) (not (<= t 1e+66)))
   (- y (* (/ (- y x) t) (- z a)))
   (+ x (/ (* (- y x) (- z t)) (- a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.85e+122) || !(t <= 1e+66)) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = x + (((y - x) * (z - t)) / (a - t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-1.85d+122)) .or. (.not. (t <= 1d+66))) then
        tmp = y - (((y - x) / t) * (z - a))
    else
        tmp = x + (((y - x) * (z - t)) / (a - t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.85e+122) || !(t <= 1e+66)) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = x + (((y - x) * (z - t)) / (a - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -1.85e+122) or not (t <= 1e+66):
		tmp = y - (((y - x) / t) * (z - a))
	else:
		tmp = x + (((y - x) * (z - t)) / (a - t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.85e+122) || !(t <= 1e+66))
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -1.85e+122) || ~((t <= 1e+66)))
		tmp = y - (((y - x) / t) * (z - a));
	else
		tmp = x + (((y - x) * (z - t)) / (a - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.85e+122], N[Not[LessEqual[t, 1e+66]], $MachinePrecision]], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8499999999999998e122 or 9.99999999999999945e65 < t

   1. Initial program 34.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 12.3

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

   8. Applied rewrites 12.3%

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

   9. Taylor expanded in t around inf

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

       1. fp-cancel-sub-sign-inv N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. associate-+l- N/A

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

       8. div-sub N/A

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

       9. lower--.f64 N/A

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

       10. div-sub N/A

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

       11. associate-/l* N/A

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

       12. associate-/l* N/A

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

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

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 83.2

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

   11. Applied rewrites 83.2%

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

   if -1.8499999999999998e122 < t < 9.99999999999999945e65

   1. Initial program 86.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+122} \lor \neg \left(t \leq 10^{+66}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+14} \lor \neg \left(t \leq 2.8 \cdot 10^{+65}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.12e+14) (not (<= t 2.8e+65)))
   (- y (* (/ (- y x) t) (- z a)))
   (fma (- y x) (/ (- z t) a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.12e+14) || !(t <= 2.8e+65)) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = fma((y - x), ((z - t) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.12e+14) || !(t <= 2.8e+65))
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	else
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.12e+14], N[Not[LessEqual[t, 2.8e+65]], $MachinePrecision]], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.12e14 or 2.7999999999999999e65 < t

   1. Initial program 43.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 20.5

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

   8. Applied rewrites 20.5%

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

   9. Taylor expanded in t around inf

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

       1. fp-cancel-sub-sign-inv N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. associate-+l- N/A

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

       8. div-sub N/A

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

       9. lower--.f64 N/A

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

       10. div-sub N/A

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

       11. associate-/l* N/A

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

       12. associate-/l* N/A

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

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

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 77.1

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

   11. Applied rewrites 77.1%

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

   if -1.12e14 < t < 2.7999999999999999e65

   1. Initial program 89.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 79.2

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

   8. Applied rewrites 79.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+14} \lor \neg \left(t \leq 2.8 \cdot 10^{+65}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(-x\right) \cdot \frac{z - a}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.12e+14)
   (fma (/ (- x y) t) z y)
   (if (<= t 4.3e+65)
     (fma (- y x) (/ (- z t) a) x)
     (- y (* (- x) (/ (- z a) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.12e+14) {
		tmp = fma(((x - y) / t), z, y);
	} else if (t <= 4.3e+65) {
		tmp = fma((y - x), ((z - t) / a), x);
	} else {
		tmp = y - (-x * ((z - a) / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.12e+14)
		tmp = fma(Float64(Float64(x - y) / t), z, y);
	elseif (t <= 4.3e+65)
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(y - Float64(Float64(-x) * Float64(Float64(z - a) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.12e+14], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z + y), $MachinePrecision], If[LessEqual[t, 4.3e+65], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y - N[((-x) * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.12e14

   1. Initial program 49.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 26.0

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

   8. Applied rewrites 26.0%

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

   9. Taylor expanded in t around inf

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

       1. fp-cancel-sub-sign-inv N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. associate-+l- N/A

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

       8. div-sub N/A

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

       9. lower--.f64 N/A

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

       10. div-sub N/A

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

       11. associate-/l* N/A

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

       12. associate-/l* N/A

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

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

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 70.9

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

   11. Applied rewrites 70.9%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 70.4%

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

   if -1.12e14 < t < 4.30000000000000046e65

   1. Initial program 89.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 79.2

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

   8. Applied rewrites 79.2%

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

   if 4.30000000000000046e65 < t

   1. Initial program 35.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 13.8

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

   8. Applied rewrites 13.8%

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

   9. Taylor expanded in t around inf

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

       1. fp-cancel-sub-sign-inv N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. associate-+l- N/A

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

       8. div-sub N/A

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

       9. lower--.f64 N/A

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

       10. div-sub N/A

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

       11. associate-/l* N/A

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

       12. associate-/l* N/A

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

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

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 84.7

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

   11. Applied rewrites 84.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 82.0%

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

Alternative 6: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-63} \lor \neg \left(a \leq 3.4\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.5e-63) (not (<= a 3.4)))
   (fma (- y x) (/ (- z t) a) x)
   (fma (/ (- x y) t) z y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.5e-63) || !(a <= 3.4)) {
		tmp = fma((y - x), ((z - t) / a), x);
	} else {
		tmp = fma(((x - y) / t), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.5e-63) || !(a <= 3.4))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / a), x);
	else
		tmp = fma(Float64(Float64(x - y) / t), z, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.5e-63], N[Not[LessEqual[a, 3.4]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e-63 or 3.39999999999999991 < a

   1. Initial program 73.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.2

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

   8. Applied rewrites 75.2%

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

   if -4.5e-63 < a < 3.39999999999999991

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 25.7

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in t around inf

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

       1. fp-cancel-sub-sign-inv N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. associate-+l- N/A

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

       8. div-sub N/A

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

       9. lower--.f64 N/A

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

       10. div-sub N/A

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

       11. associate-/l* N/A

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

       12. associate-/l* N/A

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

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

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 80.2

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

   11. Applied rewrites 80.2%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 77.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-63} \lor \neg \left(a \leq 3.4\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+33}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.3e+33)
   y
   (if (<= t 1.28e-288)
     (fma (- x) (/ z a) x)
     (if (<= t 3.1e+86) (fma (/ y a) z x) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.3e+33) {
		tmp = y;
	} else if (t <= 1.28e-288) {
		tmp = fma(-x, (z / a), x);
	} else if (t <= 3.1e+86) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.3e+33)
		tmp = y;
	elseif (t <= 1.28e-288)
		tmp = fma(Float64(-x), Float64(z / a), x);
	elseif (t <= 3.1e+86)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.3e+33], y, If[LessEqual[t, 1.28e-288], N[((-x) * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 3.1e+86], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], y]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.30000000000000011e33 or 3.1000000000000002e86 < t

   1. Initial program 39.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 33.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 33.6%

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

   13. Applied rewrites 48.7%

       \[\leadsto y + 0 \]

   if -2.30000000000000011e33 < t < 1.2800000000000001e-288

   1. Initial program 92.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-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 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.0%

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

   if 1.2800000000000001e-288 < t < 3.1000000000000002e86

   1. Initial program 84.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   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 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.7%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+33}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.5e-63)
   (fma (- y x) (/ (- z t) a) x)
   (if (<= a 3.4) (fma (/ (- x y) t) z y) (fma (- z t) (/ (- y x) a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.5e-63) {
		tmp = fma((y - x), ((z - t) / a), x);
	} else if (a <= 3.4) {
		tmp = fma(((x - y) / t), z, y);
	} else {
		tmp = fma((z - t), ((y - x) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.5e-63)
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / a), x);
	elseif (a <= 3.4)
		tmp = fma(Float64(Float64(x - y) / t), z, y);
	else
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.5e-63], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 3.4], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z + y), $MachinePrecision], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.5e-63

   1. Initial program 74.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 78.1

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

   8. Applied rewrites 78.1%

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

   if -4.5e-63 < a < 3.39999999999999991

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 25.7

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in t around inf

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

       1. fp-cancel-sub-sign-inv N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. associate-+l- N/A

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

       8. div-sub N/A

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

       9. lower--.f64 N/A

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

       10. div-sub N/A

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

       11. associate-/l* N/A

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

       12. associate-/l* N/A

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

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

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 80.2

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

   11. Applied rewrites 80.2%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 77.4%

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

   if 3.39999999999999991 < a

   1. Initial program 72.0%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 72.8

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

   5. Applied rewrites 72.8%

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

Alternative 9: 38.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{a}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-60}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+187}:\\ \;\;\;\;x - \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ z a))))
   (if (<= z -1.15e+42)
     t_1
     (if (<= z -1.35e-60) y (if (<= z 3.6e+187) (- x (- y)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / a);
	double tmp;
	if (z <= -1.15e+42) {
		tmp = t_1;
	} else if (z <= -1.35e-60) {
		tmp = y;
	} else if (z <= 3.6e+187) {
		tmp = x - -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 * (z / a)
    if (z <= (-1.15d+42)) then
        tmp = t_1
    else if (z <= (-1.35d-60)) then
        tmp = y
    else if (z <= 3.6d+187) then
        tmp = x - -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 * (z / a);
	double tmp;
	if (z <= -1.15e+42) {
		tmp = t_1;
	} else if (z <= -1.35e-60) {
		tmp = y;
	} else if (z <= 3.6e+187) {
		tmp = x - -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (z / a)
	tmp = 0
	if z <= -1.15e+42:
		tmp = t_1
	elif z <= -1.35e-60:
		tmp = y
	elif z <= 3.6e+187:
		tmp = x - -y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z / a))
	tmp = 0.0
	if (z <= -1.15e+42)
		tmp = t_1;
	elseif (z <= -1.35e-60)
		tmp = y;
	elseif (z <= 3.6e+187)
		tmp = Float64(x - Float64(-y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (z / a);
	tmp = 0.0;
	if (z <= -1.15e+42)
		tmp = t_1;
	elseif (z <= -1.35e-60)
		tmp = y;
	elseif (z <= 3.6e+187)
		tmp = x - -y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e+42], t$95$1, If[LessEqual[z, -1.35e-60], y, If[LessEqual[z, 3.6e+187], N[(x - (-y)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15e42 or 3.60000000000000036e187 < z

   1. Initial program 73.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-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 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 33.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 35.8%

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

   if -1.15e42 < z < -1.35e-60

   1. Initial program 51.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 39.4%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 39.4%

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

   13. Applied rewrites 60.4%

       \[\leadsto y + 0 \]

   if -1.35e-60 < z < 3.60000000000000036e187

   1. Initial program 67.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 33.5

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

   5. Applied rewrites 33.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 23.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 23.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 41.4%

       \[\leadsto x - \left(-y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-60}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+187}:\\ \;\;\;\;x - \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+14} \lor \neg \left(t \leq 2.8 \cdot 10^{+65}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.12e+14) (not (<= t 2.8e+65)))
   (fma (/ (- x y) t) z y)
   (fma (/ z a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.12e+14) || !(t <= 2.8e+65)) {
		tmp = fma(((x - y) / t), z, y);
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.12e+14) || !(t <= 2.8e+65))
		tmp = fma(Float64(Float64(x - y) / t), z, y);
	else
		tmp = fma(Float64(z / a), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.12e+14], N[Not[LessEqual[t, 2.8e+65]], $MachinePrecision]], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z + y), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.12e14 or 2.7999999999999999e65 < t

   1. Initial program 43.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 20.5

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

   8. Applied rewrites 20.5%

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

   9. Taylor expanded in t around inf

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

       1. fp-cancel-sub-sign-inv N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. associate-+l- N/A

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

       8. div-sub N/A

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

       9. lower--.f64 N/A

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

       10. div-sub N/A

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

       11. associate-/l* N/A

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

       12. associate-/l* N/A

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

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

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 77.1

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

   11. Applied rewrites 77.1%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 72.3%

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

   if -1.12e14 < t < 2.7999999999999999e65

   1. Initial program 89.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-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 73.6

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

   5. Applied rewrites 73.6%

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

   7. Applied rewrites 74.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+14} \lor \neg \left(t \leq 2.8 \cdot 10^{+65}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-27} \lor \neg \left(a \leq 4.7 \cdot 10^{+129}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.1e-27) (not (<= a 4.7e+129)))
   (fma (/ y a) z x)
   (fma (/ (- x y) t) z y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.1e-27) || !(a <= 4.7e+129)) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = fma(((x - y) / t), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.1e-27) || !(a <= 4.7e+129))
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = fma(Float64(Float64(x - y) / t), z, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.1e-27], N[Not[LessEqual[a, 4.7e+129]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.10000000000000015e-27 or 4.70000000000000008e129 < a

   1. Initial program 75.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      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 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.2%

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

   if -2.10000000000000015e-27 < a < 4.70000000000000008e129

   1. Initial program 63.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 31.8

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

   8. Applied rewrites 31.8%

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

   9. Taylor expanded in t around inf

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

       1. fp-cancel-sub-sign-inv N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. associate-+l- N/A

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

       8. div-sub N/A

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

       9. lower--.f64 N/A

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

       10. div-sub N/A

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

       11. associate-/l* N/A

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

       12. associate-/l* N/A

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

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

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower--.f64 74.5

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

   11. Applied rewrites 74.5%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 71.8%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-27} \lor \neg \left(a \leq 4.7 \cdot 10^{+129}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-30} \lor \neg \left(a \leq 4.7 \cdot 10^{+129}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{z}{t}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.3e-30) (not (<= a 4.7e+129)))
   (fma (/ y a) z x)
   (* (- 1.0 (/ z t)) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.3e-30) || !(a <= 4.7e+129)) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = (1.0 - (z / t)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.3e-30) || !(a <= 4.7e+129))
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(Float64(1.0 - Float64(z / t)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.3e-30], N[Not[LessEqual[a, 4.7e+129]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.29999999999999984e-30 or 4.70000000000000008e129 < a

   1. Initial program 75.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      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 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.2%

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

   if -2.29999999999999984e-30 < a < 4.70000000000000008e129

   1. Initial program 63.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 24.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.6%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-30} \lor \neg \left(a \leq 4.7 \cdot 10^{+129}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{z}{t}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{+122} \lor \neg \left(t \leq 3.1 \cdot 10^{+86}\right):\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.35e+122) (not (<= t 3.1e+86))) y (fma (/ y a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.35e+122) || !(t <= 3.1e+86)) {
		tmp = y;
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.35e+122) || !(t <= 3.1e+86))
		tmp = y;
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.35e+122], N[Not[LessEqual[t, 3.1e+86]], $MachinePrecision]], y, N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.35000000000000012e122 or 3.1000000000000002e86 < t

   1. Initial program 34.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 35.4%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 35.4%

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

   13. Applied rewrites 53.1%

       \[\leadsto y + 0 \]

   if -2.35000000000000012e122 < t < 3.1000000000000002e86

   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 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 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.6%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{+122} \lor \neg \left(t \leq 3.1 \cdot 10^{+86}\right):\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.52 \cdot 10^{-30} \lor \neg \left(a \leq 8.5 \cdot 10^{-48}\right):\\ \;\;\;\;x - \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.52e-30) (not (<= a 8.5e-48))) (- x (- y)) y))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.52e-30) || !(a <= 8.5e-48)) {
		tmp = x - -y;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.52d-30)) .or. (.not. (a <= 8.5d-48))) then
        tmp = x - -y
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.52e-30) || !(a <= 8.5e-48)) {
		tmp = x - -y;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.52e-30) or not (a <= 8.5e-48):
		tmp = x - -y
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.52e-30) || !(a <= 8.5e-48))
		tmp = Float64(x - Float64(-y));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.52e-30) || ~((a <= 8.5e-48)))
		tmp = x - -y;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.52e-30], N[Not[LessEqual[a, 8.5e-48]], $MachinePrecision]], N[(x - (-y)), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if a < -1.52e-30 or 8.5000000000000004e-48 < a

   1. Initial program 72.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 24.3

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

   5. Applied rewrites 24.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 15.5%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 40.1%

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

   if -1.52e-30 < a < 8.5000000000000004e-48

   1. Initial program 62.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 22.0%

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

   13. Applied rewrites 34.8%

       \[\leadsto y + 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.52 \cdot 10^{-30} \lor \neg \left(a \leq 8.5 \cdot 10^{-48}\right):\\ \;\;\;\;x - \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.8% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. mul-1-neg N/A

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

   11. *-lft-identity N/A

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

   12. metadata-eval N/A

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

   13. fp-cancel-sign-sub-inv N/A

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

   14. distribute-neg-in N/A

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

   15. mul-1-neg N/A

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

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

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

   17. metadata-eval N/A

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

   18. *-lft-identity N/A

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

   19. lower-fma.f64 38.3

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

5. Applied rewrites 38.3%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 18.4%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 18.4%

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

13. Applied rewrites 25.1%

    \[\leadsto y + 0 \]

14. Final simplification 25.1%

    \[\leadsto y \]
15. 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 2024338 
(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))))