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

Percentage Accurate: 68.4% → 90.7%

Time: 13.3s

Alternatives: 20

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-307} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\mathsf{fma}\left(-x, z - a, \left(z - a\right) \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (or (<= t_1 -1e-307) (not (<= t_1 0.0)))
     (fma (/ (- z t) (- a t)) (- y x) x)
     (- y (/ (fma (- x) (- z a) (* (- z a) y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if ((t_1 <= -1e-307) || !(t_1 <= 0.0)) {
		tmp = fma(((z - t) / (a - t)), (y - x), x);
	} else {
		tmp = y - (fma(-x, (z - a), ((z - a) * y)) / t);
	}
	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)))
	tmp = 0.0
	if ((t_1 <= -1e-307) || !(t_1 <= 0.0))
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x);
	else
		tmp = Float64(y - Float64(fma(Float64(-x), Float64(z - a), Float64(Float64(z - a) * y)) / t));
	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]}, If[Or[LessEqual[t$95$1, -1e-307], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[((-x) * N[(z - a), $MachinePrecision] + N[(N[(z - a), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -9.99999999999999909e-308 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 74.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 92.4

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

   4. Applied rewrites 92.4%

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

   if -9.99999999999999909e-308 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 4.3

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

   4. Applied rewrites 4.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 94.7

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -1 \cdot 10^{-307} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\mathsf{fma}\left(-x, z - a, \left(z - a\right) \cdot y\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-307} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (or (<= t_1 -1e-307) (not (<= t_1 0.0)))
     (fma (/ (- z t) (- a t)) (- y x) x)
     (- y (* (/ (- y x) t) (- z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if ((t_1 <= -1e-307) || !(t_1 <= 0.0)) {
		tmp = fma(((z - t) / (a - t)), (y - x), x);
	} else {
		tmp = y - (((y - x) / t) * (z - a));
	}
	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)))
	tmp = 0.0
	if ((t_1 <= -1e-307) || !(t_1 <= 0.0))
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x);
	else
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	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]}, If[Or[LessEqual[t$95$1, -1e-307], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -9.99999999999999909e-308 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 74.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 92.4

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

   4. Applied rewrites 92.4%

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

   if -9.99999999999999909e-308 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 4.3

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

   4. Applied rewrites 4.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 94.7

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

   7. Applied rewrites 94.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -1 \cdot 10^{-307} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{z}{a - t}\\ t_2 := \left(-y\right) \cdot \frac{z - t}{t}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{+121}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-180}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- y x) (/ z (- a t)))) (t_2 (* (- y) (/ (- z t) t))))
   (if (<= t -1.55e+121)
     t_2
     (if (<= t -1500000.0)
       t_1
       (if (<= t 8.5e-180)
         (fma (/ z a) (- y x) x)
         (if (<= t 5.5e+104) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) * (z / (a - t));
	double t_2 = -y * ((z - t) / t);
	double tmp;
	if (t <= -1.55e+121) {
		tmp = t_2;
	} else if (t <= -1500000.0) {
		tmp = t_1;
	} else if (t <= 8.5e-180) {
		tmp = fma((z / a), (y - x), x);
	} else if (t <= 5.5e+104) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) * Float64(z / Float64(a - t)))
	t_2 = Float64(Float64(-y) * Float64(Float64(z - t) / t))
	tmp = 0.0
	if (t <= -1.55e+121)
		tmp = t_2;
	elseif (t <= -1500000.0)
		tmp = t_1;
	elseif (t <= 8.5e-180)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	elseif (t <= 5.5e+104)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-y) * N[(N[(z - t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.55e+121], t$95$2, If[LessEqual[t, -1500000.0], t$95$1, If[LessEqual[t, 8.5e-180], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 5.5e+104], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.55000000000000004e121 or 5.50000000000000017e104 < t

   1. Initial program 33.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 67.9

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 38.8

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

   7. Applied rewrites 38.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 65.4%

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

   if -1.55000000000000004e121 < t < -1.5e6 or 8.4999999999999993e-180 < t < 5.50000000000000017e104

   1. Initial program 81.0%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if -1.5e6 < t < 8.4999999999999993e-180

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+121}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \mathbf{elif}\;t \leq -1500000:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-180}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+104}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+121}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \mathbf{elif}\;t \leq -1500000:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.55e+121)
   (* (- y) (/ (- z t) t))
   (if (<= t -1500000.0)
     (* (- y x) (/ z (- a t)))
     (if (<= t 4.8e-28) (fma (/ z a) (- y x) x) (* (- z t) (/ y (- a t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.55e+121) {
		tmp = -y * ((z - t) / t);
	} else if (t <= -1500000.0) {
		tmp = (y - x) * (z / (a - t));
	} else if (t <= 4.8e-28) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = (z - t) * (y / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.55e+121)
		tmp = Float64(Float64(-y) * Float64(Float64(z - t) / t));
	elseif (t <= -1500000.0)
		tmp = Float64(Float64(y - x) * Float64(z / Float64(a - t)));
	elseif (t <= 4.8e-28)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = Float64(Float64(z - t) * Float64(y / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.55e+121], N[((-y) * N[(N[(z - t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1500000.0], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.8e-28], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(z - t), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.55000000000000004e121

   1. Initial program 38.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 43.5

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

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 67.7%

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

   if -1.55000000000000004e121 < t < -1.5e6

   1. Initial program 74.3%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 56.3

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

   5. Applied rewrites 56.3%

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

   if -1.5e6 < t < 4.8000000000000004e-28

   1. Initial program 92.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 77.0

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

   7. Applied rewrites 77.0%

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

   if 4.8000000000000004e-28 < t

   1. Initial program 48.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 62.3

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

   5. Applied rewrites 62.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+121}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \mathbf{elif}\;t \leq -1500000:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+117} \lor \neg \left(t \leq 2 \cdot 10^{+30}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -9.5e+117) (not (<= t 2e+30)))
   (- y (* (/ (- y x) t) (- z a)))
   (+ x (/ (* (- y x) z) (- a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -9.5e+117) || !(t <= 2e+30)) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = x + (((y - x) * z) / (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 <= (-9.5d+117)) .or. (.not. (t <= 2d+30))) then
        tmp = y - (((y - x) / t) * (z - a))
    else
        tmp = x + (((y - x) * z) / (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 <= -9.5e+117) || !(t <= 2e+30)) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = x + (((y - x) * z) / (a - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -9.5e+117) or not (t <= 2e+30):
		tmp = y - (((y - x) / t) * (z - a))
	else:
		tmp = x + (((y - x) * z) / (a - t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -9.5e+117) || !(t <= 2e+30))
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(y - x) * z) / Float64(a - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -9.5e+117) || ~((t <= 2e+30)))
		tmp = y - (((y - x) / t) * (z - a));
	else
		tmp = x + (((y - x) * z) / (a - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -9.5e+117], N[Not[LessEqual[t, 2e+30]], $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] * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.50000000000000041e117 or 2e30 < t

   1. Initial program 41.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 84.4

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

   7. Applied rewrites 84.4%

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

   if -9.50000000000000041e117 < t < 2e30

   1. Initial program 89.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+117} \lor \neg \left(t \leq 2 \cdot 10^{+30}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.32 \lor \neg \left(t \leq 3.1 \cdot 10^{-24}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -0.32) (not (<= t 3.1e-24)))
   (- y (* (/ (- y x) t) (- z a)))
   (fma (/ (- z t) a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -0.32) || !(t <= 3.1e-24)) {
		tmp = y - (((y - x) / t) * (z - a));
	} else {
		tmp = fma(((z - t) / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -0.32) || !(t <= 3.1e-24))
		tmp = Float64(y - Float64(Float64(Float64(y - x) / t) * Float64(z - a)));
	else
		tmp = fma(Float64(Float64(z - t) / a), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -0.32], N[Not[LessEqual[t, 3.1e-24]], $MachinePrecision]], N[(y - N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.320000000000000007 or 3.1e-24 < t

   1. Initial program 49.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 75.4

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 78.6

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

   7. Applied rewrites 78.6%

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

   if -0.320000000000000007 < t < 3.1e-24

   1. Initial program 92.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 79.9

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

   7. Applied rewrites 79.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.32 \lor \neg \left(t \leq 3.1 \cdot 10^{-24}\right):\\ \;\;\;\;y - \frac{y - x}{t} \cdot \left(z - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{x} \cdot x\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4600:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y x) x)))
   (if (<= t -4.2e+118)
     t_1
     (if (<= t 4600.0)
       (fma (/ (- y x) a) z x)
       (if (<= t 6.5e+118) (* x (/ (- z a) t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / x) * x;
	double tmp;
	if (t <= -4.2e+118) {
		tmp = t_1;
	} else if (t <= 4600.0) {
		tmp = fma(((y - x) / a), z, x);
	} else if (t <= 6.5e+118) {
		tmp = x * ((z - a) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / x) * x)
	tmp = 0.0
	if (t <= -4.2e+118)
		tmp = t_1;
	elseif (t <= 4600.0)
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	elseif (t <= 6.5e+118)
		tmp = Float64(x * Float64(Float64(z - a) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -4.2e+118], t$95$1, If[LessEqual[t, 4600.0], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t, 6.5e+118], N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.2e118 or 6.5e118 < t

   1. Initial program 33.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 56.2%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{y}{x} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 51.6%

      \[\leadsto \frac{y}{x} \cdot x \]

   if -4.2e118 < t < 4600

   1. Initial program 90.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

   if 4600 < t < 6.5e118

   1. Initial program 73.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 71.8

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

   7. Applied rewrites 71.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 38.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+118}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{elif}\;t \leq 4600:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{x} \cdot x\\ \mathbf{if}\;t \leq -8 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y x) x)))
   (if (<= t -8e+119)
     t_1
     (if (<= t 3000.0)
       (fma (/ y a) z x)
       (if (<= t 6.5e+118) (* x (/ (- z a) t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / x) * x;
	double tmp;
	if (t <= -8e+119) {
		tmp = t_1;
	} else if (t <= 3000.0) {
		tmp = fma((y / a), z, x);
	} else if (t <= 6.5e+118) {
		tmp = x * ((z - a) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / x) * x)
	tmp = 0.0
	if (t <= -8e+119)
		tmp = t_1;
	elseif (t <= 3000.0)
		tmp = fma(Float64(y / a), z, x);
	elseif (t <= 6.5e+118)
		tmp = Float64(x * Float64(Float64(z - a) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -8e+119], t$95$1, If[LessEqual[t, 3000.0], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t, 6.5e+118], N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.99999999999999955e119 or 6.5e118 < t

   1. Initial program 33.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 56.8%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{y}{x} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 52.2%

      \[\leadsto \frac{y}{x} \cdot x \]

   if -7.99999999999999955e119 < t < 3e3

   1. Initial program 89.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 67.1

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

   5. Applied rewrites 67.1%

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

   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 62.5%

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

   if 3e3 < t < 6.5e118

   1. Initial program 73.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 71.8

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

   7. Applied rewrites 71.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 38.4%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+119}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{elif}\;t \leq 3000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+18} \lor \neg \left(a \leq 5.2 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.2e+18) (not (<= a 5.2e+36)))
   (fma (/ (- z t) a) (- y x) x)
   (- y (/ (* (- y x) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.2e+18) || !(a <= 5.2e+36)) {
		tmp = fma(((z - t) / a), (y - x), x);
	} else {
		tmp = y - (((y - x) * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.2e+18) || !(a <= 5.2e+36))
		tmp = fma(Float64(Float64(z - t) / a), Float64(y - x), x);
	else
		tmp = Float64(y - Float64(Float64(Float64(y - x) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.2e+18], N[Not[LessEqual[a, 5.2e+36]], $MachinePrecision]], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.2e18 or 5.2000000000000003e36 < a

   1. Initial program 70.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if -6.2e18 < a < 5.2000000000000003e36

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 78.0

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 74.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+18} \lor \neg \left(a \leq 5.2 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+18} \lor \neg \left(a \leq 5.2 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.2e+18) (not (<= a 5.2e+36)))
   (fma (- z t) (/ (- y x) a) x)
   (- y (/ (* (- y x) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.2e+18) || !(a <= 5.2e+36)) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = y - (((y - x) * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.2e+18) || !(a <= 5.2e+36))
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = Float64(y - Float64(Float64(Float64(y - x) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.2e+18], N[Not[LessEqual[a, 5.2e+36]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.2e18 or 5.2000000000000003e36 < a

   1. Initial program 70.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 76.3

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

   5. Applied rewrites 76.3%

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

   if -6.2e18 < a < 5.2000000000000003e36

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 78.0

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 74.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+18} \lor \neg \left(a \leq 5.2 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+19} \lor \neg \left(a \leq 5.2 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.8e+19) (not (<= a 5.2e+36)))
   (fma (/ z a) (- y x) x)
   (- y (/ (* (- y x) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.8e+19) || !(a <= 5.2e+36)) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = y - (((y - x) * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.8e+19) || !(a <= 5.2e+36))
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = Float64(y - Float64(Float64(Float64(y - x) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.8e+19], N[Not[LessEqual[a, 5.2e+36]], $MachinePrecision]], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.8e19 or 5.2000000000000003e36 < a

   1. Initial program 70.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 70.2

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

   7. Applied rewrites 70.2%

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

   if -4.8e19 < a < 5.2000000000000003e36

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 78.0

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 74.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+19} \lor \neg \left(a \leq 5.2 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{x} \cdot x\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-303}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y x) x)))
   (if (<= t -4.8e+118)
     t_1
     (if (<= t 4.6e-303) (* 1.0 x) (if (<= t 2.5e+29) (* y (/ z a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / x) * x;
	double tmp;
	if (t <= -4.8e+118) {
		tmp = t_1;
	} else if (t <= 4.6e-303) {
		tmp = 1.0 * x;
	} else if (t <= 2.5e+29) {
		tmp = y * (z / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y / x) * x
    if (t <= (-4.8d+118)) then
        tmp = t_1
    else if (t <= 4.6d-303) then
        tmp = 1.0d0 * x
    else if (t <= 2.5d+29) then
        tmp = y * (z / a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / x) * x;
	double tmp;
	if (t <= -4.8e+118) {
		tmp = t_1;
	} else if (t <= 4.6e-303) {
		tmp = 1.0 * x;
	} else if (t <= 2.5e+29) {
		tmp = y * (z / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / x) * x
	tmp = 0
	if t <= -4.8e+118:
		tmp = t_1
	elif t <= 4.6e-303:
		tmp = 1.0 * x
	elif t <= 2.5e+29:
		tmp = y * (z / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / x) * x)
	tmp = 0.0
	if (t <= -4.8e+118)
		tmp = t_1;
	elseif (t <= 4.6e-303)
		tmp = Float64(1.0 * x);
	elseif (t <= 2.5e+29)
		tmp = Float64(y * Float64(z / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / x) * x;
	tmp = 0.0;
	if (t <= -4.8e+118)
		tmp = t_1;
	elseif (t <= 4.6e-303)
		tmp = 1.0 * x;
	elseif (t <= 2.5e+29)
		tmp = y * (z / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -4.8e+118], t$95$1, If[LessEqual[t, 4.6e-303], N[(1.0 * x), $MachinePrecision], If[LessEqual[t, 2.5e+29], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.8e118 or 2.5e29 < t

   1. Initial program 41.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 60.1%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{y}{x} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 45.1%

      \[\leadsto \frac{y}{x} \cdot x \]

   if -4.8e118 < t < 4.59999999999999991e-303

   1. Initial program 91.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 45.3%

       \[\leadsto 1 \cdot x \]

   if 4.59999999999999991e-303 < t < 2.5e29

   1. Initial program 87.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 65.3

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 25.1%

       \[\leadsto 1 \cdot x \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 42.7%

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

Alternative 13: 32.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-303}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- y x) x)))
   (if (<= t -6.5e+118)
     t_1
     (if (<= t 4.6e-303) (* 1.0 x) (if (<= t 4.1e+29) (* y (/ z a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, (y - x), x);
	double tmp;
	if (t <= -6.5e+118) {
		tmp = t_1;
	} else if (t <= 4.6e-303) {
		tmp = 1.0 * x;
	} else if (t <= 4.1e+29) {
		tmp = y * (z / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(y - x), x)
	tmp = 0.0
	if (t <= -6.5e+118)
		tmp = t_1;
	elseif (t <= 4.6e-303)
		tmp = Float64(1.0 * x);
	elseif (t <= 4.1e+29)
		tmp = Float64(y * Float64(z / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -6.5e+118], t$95$1, If[LessEqual[t, 4.6e-303], N[(1.0 * x), $MachinePrecision], If[LessEqual[t, 4.1e+29], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.5e118 or 4.1000000000000003e29 < t

   1. Initial program 41.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 38.0%

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

   if -6.5e118 < t < 4.59999999999999991e-303

   1. Initial program 91.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 45.3%

       \[\leadsto 1 \cdot x \]

   if 4.59999999999999991e-303 < t < 4.1000000000000003e29

   1. Initial program 87.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 65.3

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 25.1%

       \[\leadsto 1 \cdot x \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 42.7%

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

Alternative 14: 62.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+118} \lor \neg \left(t \leq 1.2 \cdot 10^{-23}\right):\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3e+118) (not (<= t 1.2e-23)))
   (* (- y) (/ (- z t) t))
   (fma (/ z a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3e+118) || !(t <= 1.2e-23)) {
		tmp = -y * ((z - t) / t);
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3e+118) || !(t <= 1.2e-23))
		tmp = Float64(Float64(-y) * Float64(Float64(z - t) / t));
	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, -3e+118], N[Not[LessEqual[t, 1.2e-23]], $MachinePrecision]], N[((-y) * N[(N[(z - t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3e118 or 1.19999999999999998e-23 < t

   1. Initial program 44.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 60.6%

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

   if -3e118 < t < 1.19999999999999998e-23

   1. Initial program 90.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 71.4

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

   7. Applied rewrites 71.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+118} \lor \neg \left(t \leq 1.2 \cdot 10^{-23}\right):\\ \;\;\;\;\left(-y\right) \cdot \frac{z - t}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+118} \lor \neg \left(t \leq 2.05 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -6.5e+118) (not (<= t 2.05e+30)))
   (fma a (/ (- y x) t) y)
   (fma (/ z a) (- y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -6.5e+118) || !(t <= 2.05e+30)) {
		tmp = fma(a, ((y - x) / t), y);
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -6.5e+118) || !(t <= 2.05e+30))
		tmp = fma(a, Float64(Float64(y - x) / t), 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, -6.5e+118], N[Not[LessEqual[t, 2.05e+30]], $MachinePrecision]], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5e118 or 2.05000000000000003e30 < t

   1. Initial program 41.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 84.4

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 58.3%

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

   if -6.5e118 < t < 2.05000000000000003e30

   1. Initial program 89.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 69.4

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

   7. Applied rewrites 69.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+118} \lor \neg \left(t \leq 2.05 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+118} \lor \neg \left(t \leq 2.05 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3e+118) (not (<= t 2.05e+30)))
   (fma a (/ (- y x) t) y)
   (fma (/ (- y x) a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3e+118) || !(t <= 2.05e+30)) {
		tmp = fma(a, ((y - x) / t), y);
	} else {
		tmp = fma(((y - x) / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3e+118) || !(t <= 2.05e+30))
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	else
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3e+118], N[Not[LessEqual[t, 2.05e+30]], $MachinePrecision]], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3e118 or 2.05000000000000003e30 < t

   1. Initial program 41.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. 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 84.4

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 58.3%

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

   if -3e118 < t < 2.05000000000000003e30

   1. Initial program 89.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 66.0

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

   5. Applied rewrites 66.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+118} \lor \neg \left(t \leq 2.05 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+119} \lor \neg \left(t \leq 2.2 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -8e+119) (not (<= t 2.2e+30))) (* (/ y x) x) (fma (/ y a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -8e+119) || !(t <= 2.2e+30)) {
		tmp = (y / x) * x;
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -8e+119) || !(t <= 2.2e+30))
		tmp = Float64(Float64(y / x) * x);
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -8e+119], N[Not[LessEqual[t, 2.2e+30]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.99999999999999955e119 or 2.2e30 < t

   1. Initial program 41.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   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 60.6%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{y}{x} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 45.4%

      \[\leadsto \frac{y}{x} \cdot x \]

   if -7.99999999999999955e119 < t < 2.2e30

   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 65.6

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

   5. Applied rewrites 65.6%

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

   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 61.2%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+119} \lor \neg \left(t \leq 2.2 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+118} \lor \neg \left(t \leq 7.5 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -6.5e+118) (not (<= t 7.5e-12))) (fma 1.0 (- y x) x) (* 1.0 x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -6.5e+118) || !(t <= 7.5e-12)) {
		tmp = fma(1.0, (y - x), x);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -6.5e+118) || !(t <= 7.5e-12))
		tmp = fma(1.0, Float64(y - x), x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -6.5e+118], N[Not[LessEqual[t, 7.5e-12]], $MachinePrecision]], N[(1.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5e118 or 7.5e-12 < t

   1. Initial program 43.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 36.5%

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

   if -6.5e118 < t < 7.5e-12

   1. Initial program 90.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.3%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 37.7%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+118} \lor \neg \left(t \leq 7.5 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 25.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a):
	return 1.0 * x

Julia

function code(x, y, z, t, a)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 69.0%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
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 43.4

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

5. Applied rewrites 43.4%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 33.3%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 25.3%

    \[\leadsto 1 \cdot x \]
12. Add Preprocessing

Alternative 20: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 69.0%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. mul-1-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

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

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

   10. mul-1-neg N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 40.6

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

5. Applied rewrites 40.6%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 2.9%

   \[\leadsto 0 \]
9. 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 2024337 
(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))))