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

Percentage Accurate: 98.0% → 98.2%

Time: 8.5s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (y / ((t - a) / (t - z))) + 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 / ((t - a) / (t - z))) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f64 N/A

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

   8. neg-sub0 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. associate--r+ N/A

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

   13. neg-sub0 N/A

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

   14. remove-double-neg N/A

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

   15. lower--.f64 N/A

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

   16. neg-sub0 N/A

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

   17. lift--.f64 N/A

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

   18. sub-neg N/A

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

   19. +-commutative N/A

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

   20. associate--r+ N/A

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

   21. neg-sub0 N/A

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

   22. remove-double-neg N/A

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

   23. lower--.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{a - t}\\ t_2 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t\_1 \cdot y + x\\ \mathbf{elif}\;t\_2 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;t\_2 \leq 1.00000001:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ z (- a t))) (t_2 (/ (- z t) (- a t))))
   (if (<= t_2 -2e+34)
     (+ (* t_1 y) x)
     (if (<= t_2 0.8)
       (fma (/ (- z t) a) y x)
       (if (<= t_2 1.00000001) (fma (- y) (/ t (- a t)) x) (fma t_1 y x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z / (a - t);
	double t_2 = (z - t) / (a - t);
	double tmp;
	if (t_2 <= -2e+34) {
		tmp = (t_1 * y) + x;
	} else if (t_2 <= 0.8) {
		tmp = fma(((z - t) / a), y, x);
	} else if (t_2 <= 1.00000001) {
		tmp = fma(-y, (t / (a - t)), x);
	} else {
		tmp = fma(t_1, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z / Float64(a - t))
	t_2 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_2 <= -2e+34)
		tmp = Float64(Float64(t_1 * y) + x);
	elseif (t_2 <= 0.8)
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	elseif (t_2 <= 1.00000001)
		tmp = fma(Float64(-y), Float64(t / Float64(a - t)), x);
	else
		tmp = fma(t_1, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+34], N[(N[(t$95$1 * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 0.8], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$2, 1.00000001], N[((-y) * N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t$95$1 * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.99999999999999989e34

   1. Initial program 96.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -1.99999999999999989e34 < (/.f64 (-.f64 z t) (-.f64 a t)) < 0.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.80000000000000004 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.0000000099999999

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -2 \cdot 10^{+34}:\\ \;\;\;\;\frac{z}{a - t} \cdot y + x\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 1.00000001:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 1.00000001:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z (- a t)) y x)))
   (if (<= t_1 -2e+34)
     t_2
     (if (<= t_1 0.8)
       (fma (/ (- z t) a) y x)
       (if (<= t_1 1.00000001) (fma (- y) (/ t (- a t)) x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / (a - t)), y, x);
	double tmp;
	if (t_1 <= -2e+34) {
		tmp = t_2;
	} else if (t_1 <= 0.8) {
		tmp = fma(((z - t) / a), y, x);
	} else if (t_1 <= 1.00000001) {
		tmp = fma(-y, (t / (a - t)), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / Float64(a - t)), y, x)
	tmp = 0.0
	if (t_1 <= -2e+34)
		tmp = t_2;
	elseif (t_1 <= 0.8)
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	elseif (t_1 <= 1.00000001)
		tmp = fma(Float64(-y), Float64(t / Float64(a - t)), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+34], t$95$2, If[LessEqual[t$95$1, 0.8], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1.00000001], N[((-y) * N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.99999999999999989e34 or 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if -1.99999999999999989e34 < (/.f64 (-.f64 z t) (-.f64 a t)) < 0.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.80000000000000004 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.0000000099999999

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 4: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a - t}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 1.00000001:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z (- a t)) y x)))
   (if (<= t_1 -2e+34)
     t_2
     (if (<= t_1 0.8)
       (fma (/ (- z t) a) y x)
       (if (<= t_1 1.00000001) (fma (/ (- t z) t) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / (a - t)), y, x);
	double tmp;
	if (t_1 <= -2e+34) {
		tmp = t_2;
	} else if (t_1 <= 0.8) {
		tmp = fma(((z - t) / a), y, x);
	} else if (t_1 <= 1.00000001) {
		tmp = fma(((t - z) / t), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / Float64(a - t)), y, x)
	tmp = 0.0
	if (t_1 <= -2e+34)
		tmp = t_2;
	elseif (t_1 <= 0.8)
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	elseif (t_1 <= 1.00000001)
		tmp = fma(Float64(Float64(t - z) / t), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+34], t$95$2, If[LessEqual[t$95$1, 0.8], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1.00000001], N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.99999999999999989e34 or 1.0000000099999999 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if -1.99999999999999989e34 < (/.f64 (-.f64 z t) (-.f64 a t)) < 0.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.80000000000000004 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.0000000099999999

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 98.7%

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

Alternative 5: 80.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{z \cdot y}{a} + x\\ \mathbf{elif}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+130}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -2e-92)
     (+ (/ (* z y) a) x)
     (if (<= t_1 1e-8)
       (fma (- y) (/ t a) x)
       (if (<= t_1 5e+130) (+ y x) (* (/ y (- a t)) z))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -2e-92) {
		tmp = ((z * y) / a) + x;
	} else if (t_1 <= 1e-8) {
		tmp = fma(-y, (t / a), x);
	} else if (t_1 <= 5e+130) {
		tmp = y + x;
	} else {
		tmp = (y / (a - t)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -2e-92)
		tmp = Float64(Float64(Float64(z * y) / a) + x);
	elseif (t_1 <= 1e-8)
		tmp = fma(Float64(-y), Float64(t / a), x);
	elseif (t_1 <= 5e+130)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / Float64(a - t)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-92], N[(N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-8], N[((-y) * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+130], N[(y + x), $MachinePrecision], N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.99999999999999998e-92

   1. Initial program 97.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -1.99999999999999998e-92 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 94.0%

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

   if 1e-8 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999996e130

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 4.9999999999999996e130 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{z \cdot y}{a} + x\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{+130}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+130}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -2e-68)
     (fma (/ z a) y x)
     (if (<= t_1 1e-8)
       (fma (- y) (/ t a) x)
       (if (<= t_1 5e+130) (+ y x) (* (/ y (- a t)) z))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -2e-68) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 1e-8) {
		tmp = fma(-y, (t / a), x);
	} else if (t_1 <= 5e+130) {
		tmp = y + x;
	} else {
		tmp = (y / (a - t)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -2e-68)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 1e-8)
		tmp = fma(Float64(-y), Float64(t / a), x);
	elseif (t_1 <= 5e+130)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / Float64(a - t)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-68], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-8], N[((-y) * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+130], N[(y + x), $MachinePrecision], N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -2.00000000000000013e-68

   1. Initial program 97.7%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -2.00000000000000013e-68 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 92.0%

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

   if 1e-8 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999996e130

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 4.9999999999999996e130 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

Alternative 7: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{z \cdot y}{a} + x\\ \mathbf{elif}\;t\_1 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -2e-92)
     (+ (/ (* z y) a) x)
     (if (<= t_1 0.8) (fma (- y) (/ t a) x) (fma (/ (- t z) t) y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -2e-92) {
		tmp = ((z * y) / a) + x;
	} else if (t_1 <= 0.8) {
		tmp = fma(-y, (t / a), x);
	} else {
		tmp = fma(((t - z) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -2e-92)
		tmp = Float64(Float64(Float64(z * y) / a) + x);
	elseif (t_1 <= 0.8)
		tmp = fma(Float64(-y), Float64(t / a), x);
	else
		tmp = fma(Float64(Float64(t - z) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-92], N[(N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.8], N[((-y) * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.99999999999999998e-92

   1. Initial program 97.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -1.99999999999999998e-92 < (/.f64 (-.f64 z t) (-.f64 a t)) < 0.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 93.1%

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

   if 0.80000000000000004 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 89.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{z \cdot y}{a} + x\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+130}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 0.8)
     (fma (/ z a) y x)
     (if (<= t_1 5e+130) (+ y x) (* (/ y (- a t)) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 0.8) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 5e+130) {
		tmp = y + x;
	} else {
		tmp = (y / (a - t)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 0.8)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 5e+130)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / Float64(a - t)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.8], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+130], N[(y + x), $MachinePrecision], N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 0.80000000000000004

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 0.80000000000000004 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999996e130

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if 4.9999999999999996e130 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

Alternative 9: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+113}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 0.8)
     (fma (/ z a) y x)
     (if (<= t_1 5e+113) (+ y x) (fma z (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 0.8) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 5e+113) {
		tmp = y + x;
	} else {
		tmp = fma(z, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 0.8)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 5e+113)
		tmp = Float64(y + x);
	else
		tmp = fma(z, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.8], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+113], N[(y + x), $MachinePrecision], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 0.80000000000000004

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 0.80000000000000004 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5e113

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 5e113 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 66.6

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

   7. Applied rewrites 66.6%

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

Alternative 10: 79.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{if}\;t\_1 \leq 0.8:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+113}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z a) y x)))
   (if (<= t_1 0.8) t_2 (if (<= t_1 5e+113) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / a), y, x);
	double tmp;
	if (t_1 <= 0.8) {
		tmp = t_2;
	} else if (t_1 <= 5e+113) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / a), y, x)
	tmp = 0.0
	if (t_1 <= 0.8)
		tmp = t_2;
	elseif (t_1 <= 5e+113)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, 0.8], t$95$2, If[LessEqual[t$95$1, 5e+113], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 0.80000000000000004 or 5e113 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 0.80000000000000004 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5e113

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 11: 64.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+130}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -2e+66)
     (/ (* z y) a)
     (if (<= t_1 5e+130) (+ y x) (* (/ y a) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -2e+66) {
		tmp = (z * y) / a;
	} else if (t_1 <= 5e+130) {
		tmp = y + x;
	} else {
		tmp = (y / a) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (a - t)
    if (t_1 <= (-2d+66)) then
        tmp = (z * y) / a
    else if (t_1 <= 5d+130) then
        tmp = y + x
    else
        tmp = (y / a) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -2e+66) {
		tmp = (z * y) / a;
	} else if (t_1 <= 5e+130) {
		tmp = y + x;
	} else {
		tmp = (y / a) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	tmp = 0
	if t_1 <= -2e+66:
		tmp = (z * y) / a
	elif t_1 <= 5e+130:
		tmp = y + x
	else:
		tmp = (y / a) * z
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -2e+66)
		tmp = Float64(Float64(z * y) / a);
	elseif (t_1 <= 5e+130)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / a) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	tmp = 0.0;
	if (t_1 <= -2e+66)
		tmp = (z * y) / a;
	elseif (t_1 <= 5e+130)
		tmp = y + x;
	else
		tmp = (y / a) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+66], N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 5e+130], N[(y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.99999999999999989e66

   1. Initial program 95.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 57.6%

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

   if -1.99999999999999989e66 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999996e130

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if 4.9999999999999996e130 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 49.3%

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

Alternative 12: 64.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \frac{z \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+130}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (/ (* z y) a)))
   (if (<= t_1 -2e+66) t_2 (if (<= t_1 5e+130) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (z * y) / a;
	double tmp;
	if (t_1 <= -2e+66) {
		tmp = t_2;
	} else if (t_1 <= 5e+130) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (a - t)
    t_2 = (z * y) / a
    if (t_1 <= (-2d+66)) then
        tmp = t_2
    else if (t_1 <= 5d+130) then
        tmp = y + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (z * y) / a;
	double tmp;
	if (t_1 <= -2e+66) {
		tmp = t_2;
	} else if (t_1 <= 5e+130) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	t_2 = (z * y) / a
	tmp = 0
	if t_1 <= -2e+66:
		tmp = t_2
	elif t_1 <= 5e+130:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(Float64(z * y) / a)
	tmp = 0.0
	if (t_1 <= -2e+66)
		tmp = t_2;
	elseif (t_1 <= 5e+130)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	t_2 = (z * y) / a;
	tmp = 0.0;
	if (t_1 <= -2e+66)
		tmp = t_2;
	elseif (t_1 <= 5e+130)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+66], t$95$2, If[LessEqual[t$95$1, 5e+130], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.99999999999999989e66 or 4.9999999999999996e130 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 54.0%

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

   if -1.99999999999999989e66 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999996e130

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 73.3

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

   5. Applied rewrites 73.3%

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

Alternative 13: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \frac{z}{a} \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+130}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (* (/ z a) y)))
   (if (<= t_1 -2e+66) t_2 (if (<= t_1 5e+130) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (z / a) * y;
	double tmp;
	if (t_1 <= -2e+66) {
		tmp = t_2;
	} else if (t_1 <= 5e+130) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (a - t)
    t_2 = (z / a) * y
    if (t_1 <= (-2d+66)) then
        tmp = t_2
    else if (t_1 <= 5d+130) then
        tmp = y + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (z / a) * y;
	double tmp;
	if (t_1 <= -2e+66) {
		tmp = t_2;
	} else if (t_1 <= 5e+130) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	t_2 = (z / a) * y
	tmp = 0
	if t_1 <= -2e+66:
		tmp = t_2
	elif t_1 <= 5e+130:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(Float64(z / a) * y)
	tmp = 0.0
	if (t_1 <= -2e+66)
		tmp = t_2;
	elseif (t_1 <= 5e+130)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	t_2 = (z / a) * y;
	tmp = 0.0;
	if (t_1 <= -2e+66)
		tmp = t_2;
	elseif (t_1 <= 5e+130)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+66], t$95$2, If[LessEqual[t$95$1, 5e+130], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.99999999999999989e66 or 4.9999999999999996e130 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.4%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 54.0%

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

   10. Applied rewrites 51.6%

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

   if -1.99999999999999989e66 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999996e130

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 73.3

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

   5. Applied rewrites 73.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -2 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{+130}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- a t)) 0.8)
   (fma (/ (- z t) a) y x)
   (fma (/ (- t z) t) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= 0.8) {
		tmp = fma(((z - t) / a), y, x);
	} else {
		tmp = fma(((t - z) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(a - t)) <= 0.8)
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	else
		tmp = fma(Float64(Float64(t - z) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], 0.8], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 0.80000000000000004

   1. Initial program 99.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if 0.80000000000000004 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 89.5%

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

Alternative 15: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 16: 59.5% 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 99.5%

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

3. Taylor expanded in t around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 64.8

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

5. Applied rewrites 64.8%

   \[\leadsto \color{blue}{y + x} \]
6. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (< y -8.508084860551241e-17)
     t_1
     (if (< y 2.894426862792089e-49)
       (+ x (* (* y (- z t)) (/ 1.0 (- a t))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * ((z - t) / (a - t)))
    if (y < (-8.508084860551241d-17)) then
        tmp = t_1
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) * (1.0d0 / (a - t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	tmp = 0
	if y < -8.508084860551241e-17:
		tmp = t_1
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) * Float64(1.0 / Float64(a - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -8.508084860551241e-17], t$95$1, If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024248 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8508084860551241/100000000000000000000000000000000) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t)))))))

  (+ x (* y (/ (- z t) (- a t)))))