Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A

Percentage Accurate: 97.9% → 97.9%

Time: 6.9s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[x + y \cdot \frac{z - t}{z - a} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+279}\right):\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+279)))
     (* (/ y a) t)
     (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+279)) {
		tmp = (y / a) * t;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e+279)) {
		tmp = (y / a) * t;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (z - a))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e+279):
		tmp = (y / a) * t
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+279))
		tmp = Float64(Float64(y / a) * t);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (z - a));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e+279)))
		tmp = (y / a) * t;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+279]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -inf.0 or 1.00000000000000006e279 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 93.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 68.0

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

   7. Applied rewrites 68.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 68.0%

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

   if -inf.0 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 1.00000000000000006e279

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 68.0

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{z - a} \leq -\infty \lor \neg \left(y \cdot \frac{z - t}{z - a} \leq 10^{+279}\right):\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+279}\right):\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+279)))
     (* (/ t a) y)
     (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+279)) {
		tmp = (t / a) * y;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e+279)) {
		tmp = (t / a) * y;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (z - a))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e+279):
		tmp = (t / a) * y
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+279))
		tmp = Float64(Float64(t / a) * y);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (z - a));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e+279)))
		tmp = (t / a) * y;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+279]], $MachinePrecision]], N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -inf.0 or 1.00000000000000006e279 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 93.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 68.0

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

   7. Applied rewrites 68.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.1%

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

   12. Applied rewrites 64.9%

       \[\leadsto \frac{t}{a} \cdot y \]

   if -inf.0 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 1.00000000000000006e279

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 68.0

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{z - a} \leq -\infty \lor \neg \left(y \cdot \frac{z - t}{z - a} \leq 10^{+279}\right):\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+279}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (<= t_1 (- INFINITY))
     (/ (* t y) a)
     (if (<= t_1 1e+279) (+ y x) (* (/ y a) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t * y) / a;
	} else if (t_1 <= 1e+279) {
		tmp = y + x;
	} else {
		tmp = (y / a) * t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (t * y) / a;
	} else if (t_1 <= 1e+279) {
		tmp = y + x;
	} else {
		tmp = (y / a) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (z - a))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (t * y) / a
	elif t_1 <= 1e+279:
		tmp = y + x
	else:
		tmp = (y / a) * t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t * y) / a);
	elseif (t_1 <= 1e+279)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / a) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (z - a));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (t * y) / a;
	elseif (t_1 <= 1e+279)
		tmp = y + x;
	else
		tmp = (y / a) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 1e+279], N[(y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 91.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 90.9

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.7

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

   7. Applied rewrites 81.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 81.8%

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

   if -inf.0 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 1.00000000000000006e279

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 68.0

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{y + x} \]

   if 1.00000000000000006e279 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 95.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 85.0

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 60.5

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 60.5%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{z - a} \leq -\infty:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{elif}\;y \cdot \frac{z - t}{z - a} \leq 10^{+279}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+66}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{-y}{z}, t - a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+66)
     (* (- z t) (/ y (- z a)))
     (if (<= t_1 2e-7) (fma (/ t a) y x) (+ x (fma (/ (- y) z) (- t a) y))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+66) {
		tmp = (z - t) * (y / (z - a));
	} else if (t_1 <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = x + fma((-y / z), (t - a), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+66)
		tmp = Float64(Float64(z - t) * Float64(y / Float64(z - a)));
	elseif (t_1 <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = Float64(x + fma(Float64(Float64(-y) / z), Float64(t - a), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+66], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(N[((-y) / z), $MachinePrecision] * N[(t - a), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.99999999999999991e66

   1. Initial program 93.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -4.99999999999999991e66 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 88.3

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 89.8%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

      12. distribute-frac-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+66}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{-y}{z}, t - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+66}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+66)
     (* (- z t) (/ y (- z a)))
     (if (<= t_1 2e-7) (fma (/ t a) y x) (fma y (- 1.0 (/ t z)) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+66) {
		tmp = (z - t) * (y / (z - a));
	} else if (t_1 <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = fma(y, (1.0 - (t / z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+66)
		tmp = Float64(Float64(z - t) * Float64(y / Float64(z - a)));
	elseif (t_1 <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+66], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.99999999999999991e66

   1. Initial program 93.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -4.99999999999999991e66 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 88.3

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 89.8%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 93.4

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

   7. Applied rewrites 93.4%

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

   9. Applied rewrites 93.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+66}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+66}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{z - a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+66)
     (* (- y) (/ t (- z a)))
     (if (<= t_1 2e-7) (fma (/ t a) y x) (fma y (- 1.0 (/ t z)) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+66) {
		tmp = -y * (t / (z - a));
	} else if (t_1 <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = fma(y, (1.0 - (t / z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+66)
		tmp = Float64(Float64(-y) * Float64(t / Float64(z - a)));
	elseif (t_1 <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+66], N[((-y) * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.99999999999999991e66

   1. Initial program 93.3%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{z - a}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -4.99999999999999991e66 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 88.3

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 89.8%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 93.4

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

   7. Applied rewrites 93.4%

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

   9. Applied rewrites 93.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+66}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-7)
     (fma (/ t a) y x)
     (if (<= t_1 200.0) (fma (/ z (- z a)) y x) (fma y (/ (- t) z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 200.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = fma(y, (-t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 200.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = fma(y, Float64(Float64(-t) / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.4%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a)) < 200

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 200 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 92.0

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 82.8

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

   7. Applied rewrites 82.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 81.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-7)
     (fma (/ t a) y x)
     (if (<= t_1 200.0) (+ y x) (fma y (/ (- t) z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 200.0) {
		tmp = y + x;
	} else {
		tmp = fma(y, (-t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 200.0)
		tmp = Float64(y + x);
	else
		tmp = fma(y, Float64(Float64(-t) / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(y + x), $MachinePrecision], N[(y * N[((-t) / z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.4%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a)) < 200

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 98.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{y + x} \]

   if 200 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 92.0

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 82.8

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

   7. Applied rewrites 82.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 81.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 200:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-7)
     (fma (/ t a) y x)
     (if (<= t_1 200.0) (+ y x) (- y (* y (/ t z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 200.0) {
		tmp = y + x;
	} else {
		tmp = y - (y * (t / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 200.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(y - Float64(y * Float64(t / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(y + x), $MachinePrecision], N[(y - N[(y * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.4%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a)) < 200

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 98.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{y + x} \]

   if 200 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 92.0

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 82.8

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

   7. Applied rewrites 82.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.5%

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

   11. Taylor expanded in z around inf

       \[\leadsto y + -1 \cdot \color{blue}{\frac{t \cdot y}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 66.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 200:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-7)
     (fma (/ t a) y x)
     (if (<= t_1 200.0) (+ y x) (* (- y) (/ t z))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 200.0) {
		tmp = y + x;
	} else {
		tmp = -y * (t / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 200.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(-y) * Float64(t / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(y + x), $MachinePrecision], N[((-y) * N[(t / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.4%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a)) < 200

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 98.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{y + x} \]

   if 200 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{z - a}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 65.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 200:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 2e-7)
     (fma (/ t a) y x)
     (if (<= t_1 200.0) (+ y x) (* t (/ (- y) z))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 200.0) {
		tmp = y + x;
	} else {
		tmp = t * (-y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 200.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(t * Float64(Float64(-y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(y + x), $MachinePrecision], N[(t * N[((-y) / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.4%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a)) < 200

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 98.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{y + x} \]

   if 200 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{z - a}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 65.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 200:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-7} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+152}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 2e-7) (not (<= t_1 5e+152))) (fma (/ t a) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 2e-7) || !(t_1 <= 5e+152)) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 2e-7) || !(t_1 <= 5e+152))
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 2e-7], N[Not[LessEqual[t$95$1, 5e+152]], $MachinePrecision]], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7 or 5e152 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 77.3

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.4

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

   7. Applied rewrites 80.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 80.8%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e152

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 85.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 85.6%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7} \lor \neg \left(\frac{z - t}{z - a} \leq 5 \cdot 10^{+152}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-7} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+152}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 2e-7) (not (<= t_1 5e+152))) (fma (/ y a) t x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 2e-7) || !(t_1 <= 5e+152)) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 2e-7) || !(t_1 <= 5e+152))
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 2e-7], N[Not[LessEqual[t$95$1, 5e+152]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7 or 5e152 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e152

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 85.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 85.6%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7} \lor \neg \left(\frac{z - t}{z - a} \leq 5 \cdot 10^{+152}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 2e-7)
   (fma (/ t a) y x)
   (fma y (- 1.0 (/ t z)) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = fma(y, (1.0 - (t / z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.4%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 93.4

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

   7. Applied rewrites 93.4%

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

   9. Applied rewrites 93.4%

      \[\leadsto \mathsf{fma}\left(y, 1 - \color{blue}{\frac{t}{z}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 2e-7) (fma (/ t a) y x) (fma (/ (- z t) z) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 2e-7) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = fma(((z - t) / z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 2e-7)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 2e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999999e-7

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.4%

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

   if 1.9999999999999999e-7 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. div-sub N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{y + x} \]

   2. lower-+.f64 60.6

      \[\leadsto \color{blue}{y + x} \]

5. Applied rewrites 60.6%

   \[\leadsto \color{blue}{y + x} \]

6. Final simplification 60.6%

   \[\leadsto y + x \]
7. Add Preprocessing

Developer Target 1: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{\frac{z - a}{z - t}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x + (y / ((z - a) / (z - 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 / ((z - a) / (z - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ y (/ (- z a) (- z t)))))

  (+ x (* y (/ (- z t) (- z a)))))