Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Percentage Accurate: 89.1% → 98.5%

Time: 9.2s

Alternatives: 14

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+14} \lor \neg \left(z \leq 800000\right):\\ \;\;\;\;\frac{{\left(\frac{t - \frac{x}{z}}{y}\right)}^{-1} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.5e+14) (not (<= z 800000.0)))
   (/ (+ (pow (/ (- t (/ x z)) y) -1.0) x) (+ 1.0 x))
   (/ (+ (/ (- (* y z) x) (- (* t z) x)) x) (+ 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e+14) || !(z <= 800000.0)) {
		tmp = (pow(((t - (x / z)) / y), -1.0) + x) / (1.0 + x);
	} else {
		tmp = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-5.5d+14)) .or. (.not. (z <= 800000.0d0))) then
        tmp = ((((t - (x / z)) / y) ** (-1.0d0)) + x) / (1.0d0 + x)
    else
        tmp = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e+14) || !(z <= 800000.0)) {
		tmp = (Math.pow(((t - (x / z)) / y), -1.0) + x) / (1.0 + x);
	} else {
		tmp = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.5e+14) or not (z <= 800000.0):
		tmp = (math.pow(((t - (x / z)) / y), -1.0) + x) / (1.0 + x)
	else:
		tmp = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.5e+14) || !(z <= 800000.0))
		tmp = Float64(Float64((Float64(Float64(t - Float64(x / z)) / y) ^ -1.0) + x) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.5e+14) || ~((z <= 800000.0)))
		tmp = ((((t - (x / z)) / y) ^ -1.0) + x) / (1.0 + x);
	else
		tmp = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.5e+14], N[Not[LessEqual[z, 800000.0]], $MachinePrecision]], N[(N[(N[Power[N[(N[(t - N[(x / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -1.0], $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e14 or 8e5 < z

   1. Initial program 81.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 81.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 81.1

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 85.9

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 98.3%

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

   if -5.5e14 < z < 8e5

   1. Initial program 99.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+14} \lor \neg \left(z \leq 800000\right):\\ \;\;\;\;\frac{{\left(\frac{t - \frac{x}{z}}{y}\right)}^{-1} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{\frac{z}{t\_1} \cdot y}{1 + x}\\ t_3 := \frac{\frac{y \cdot z - x}{t\_1} + x}{1 + x}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1 + x}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{1 + x}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (* (/ z t_1) y) (+ 1.0 x)))
        (t_3 (/ (+ (/ (- (* y z) x) t_1) x) (+ 1.0 x))))
   (if (<= t_3 -5e+14)
     t_2
     (if (<= t_3 2e-7)
       (/ (- x (/ (- (/ x z) y) t)) (+ 1.0 x))
       (if (<= t_3 2.0)
         (/ (- x (/ x t_1)) (+ 1.0 x))
         (if (<= t_3 INFINITY) t_2 (/ (+ (/ y t) x) (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = ((z / t_1) * y) / (1.0 + x);
	double t_3 = ((((y * z) - x) / t_1) + x) / (1.0 + x);
	double tmp;
	if (t_3 <= -5e+14) {
		tmp = t_2;
	} else if (t_3 <= 2e-7) {
		tmp = (x - (((x / z) - y) / t)) / (1.0 + x);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_1)) / (1.0 + x);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = ((z / t_1) * y) / (1.0 + x);
	double t_3 = ((((y * z) - x) / t_1) + x) / (1.0 + x);
	double tmp;
	if (t_3 <= -5e+14) {
		tmp = t_2;
	} else if (t_3 <= 2e-7) {
		tmp = (x - (((x / z) - y) / t)) / (1.0 + x);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_1)) / (1.0 + x);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = ((z / t_1) * y) / (1.0 + x)
	t_3 = ((((y * z) - x) / t_1) + x) / (1.0 + x)
	tmp = 0
	if t_3 <= -5e+14:
		tmp = t_2
	elif t_3 <= 2e-7:
		tmp = (x - (((x / z) - y) / t)) / (1.0 + x)
	elif t_3 <= 2.0:
		tmp = (x - (x / t_1)) / (1.0 + x)
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = ((y / t) + x) / (1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(Float64(z / t_1) * y) / Float64(1.0 + x))
	t_3 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / t_1) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_3 <= -5e+14)
		tmp = t_2;
	elseif (t_3 <= 2e-7)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(1.0 + x));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(1.0 + x));
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = ((z / t_1) * y) / (1.0 + x);
	t_3 = ((((y * z) - x) / t_1) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_3 <= -5e+14)
		tmp = t_2;
	elseif (t_3 <= 2e-7)
		tmp = (x - (((x / z) - y) / t)) / (1.0 + x);
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_1)) / (1.0 + x);
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = ((y / t) + x) / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z / t$95$1), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+14], t$95$2, If[LessEqual[t$95$3, 2e-7], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 84.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -5e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7

   1. Initial program 97.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 1.9999999999999999e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{z}{t \cdot z - x} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t \cdot z - x} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{1 + x}\\ t_2 := t \cdot z - x\\ t_3 := \frac{\frac{z}{t\_2} \cdot y}{1 + x}\\ t_4 := \frac{\frac{y \cdot z - x}{t\_2} + x}{1 + x}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (- (* t z) x))
        (t_3 (/ (* (/ z t_2) y) (+ 1.0 x)))
        (t_4 (/ (+ (/ (- (* y z) x) t_2) x) (+ 1.0 x))))
   (if (<= t_4 -5e+14)
     t_3
     (if (<= t_4 2e-7)
       t_1
       (if (<= t_4 2.0)
         (/ (- x (/ x t_2)) (+ 1.0 x))
         (if (<= t_4 INFINITY) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (t * z) - x;
	double t_3 = ((z / t_2) * y) / (1.0 + x);
	double t_4 = ((((y * z) - x) / t_2) + x) / (1.0 + x);
	double tmp;
	if (t_4 <= -5e+14) {
		tmp = t_3;
	} else if (t_4 <= 2e-7) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (t * z) - x;
	double t_3 = ((z / t_2) * y) / (1.0 + x);
	double t_4 = ((((y * z) - x) / t_2) + x) / (1.0 + x);
	double tmp;
	if (t_4 <= -5e+14) {
		tmp = t_3;
	} else if (t_4 <= 2e-7) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (1.0 + x)
	t_2 = (t * z) - x
	t_3 = ((z / t_2) * y) / (1.0 + x)
	t_4 = ((((y * z) - x) / t_2) + x) / (1.0 + x)
	tmp = 0
	if t_4 <= -5e+14:
		tmp = t_3
	elif t_4 <= 2e-7:
		tmp = t_1
	elif t_4 <= 2.0:
		tmp = (x - (x / t_2)) / (1.0 + x)
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(Float64(z / t_2) * y) / Float64(1.0 + x))
	t_4 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / t_2) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= -5e+14)
		tmp = t_3;
	elseif (t_4 <= 2e-7)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(1.0 + x));
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (1.0 + x);
	t_2 = (t * z) - x;
	t_3 = ((z / t_2) * y) / (1.0 + x);
	t_4 = ((((y * z) - x) / t_2) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_4 <= -5e+14)
		tmp = t_3;
	elseif (t_4 <= 2e-7)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_2)) / (1.0 + x);
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+14], t$95$3, If[LessEqual[t$95$4, 2e-7], t$95$1, If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 84.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -5e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 1.9999999999999999e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{z}{t \cdot z - x} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t \cdot z - x} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{1 + x}\\ t_2 := t \cdot z - x\\ t_3 := \frac{z}{1 + x} \cdot \frac{y}{t\_2}\\ t_4 := \frac{\frac{y \cdot z - x}{t\_2} + x}{1 + x}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (- (* t z) x))
        (t_3 (* (/ z (+ 1.0 x)) (/ y t_2)))
        (t_4 (/ (+ (/ (- (* y z) x) t_2) x) (+ 1.0 x))))
   (if (<= t_4 -5e+14)
     t_3
     (if (<= t_4 2e-7)
       t_1
       (if (<= t_4 2.0)
         (/ (- x (/ x t_2)) (+ 1.0 x))
         (if (<= t_4 INFINITY) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (t * z) - x;
	double t_3 = (z / (1.0 + x)) * (y / t_2);
	double t_4 = ((((y * z) - x) / t_2) + x) / (1.0 + x);
	double tmp;
	if (t_4 <= -5e+14) {
		tmp = t_3;
	} else if (t_4 <= 2e-7) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (t * z) - x;
	double t_3 = (z / (1.0 + x)) * (y / t_2);
	double t_4 = ((((y * z) - x) / t_2) + x) / (1.0 + x);
	double tmp;
	if (t_4 <= -5e+14) {
		tmp = t_3;
	} else if (t_4 <= 2e-7) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (1.0 + x)
	t_2 = (t * z) - x
	t_3 = (z / (1.0 + x)) * (y / t_2)
	t_4 = ((((y * z) - x) / t_2) + x) / (1.0 + x)
	tmp = 0
	if t_4 <= -5e+14:
		tmp = t_3
	elif t_4 <= 2e-7:
		tmp = t_1
	elif t_4 <= 2.0:
		tmp = (x - (x / t_2)) / (1.0 + x)
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(z / Float64(1.0 + x)) * Float64(y / t_2))
	t_4 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / t_2) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= -5e+14)
		tmp = t_3;
	elseif (t_4 <= 2e-7)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(1.0 + x));
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (1.0 + x);
	t_2 = (t * z) - x;
	t_3 = (z / (1.0 + x)) * (y / t_2);
	t_4 = ((((y * z) - x) / t_2) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_4 <= -5e+14)
		tmp = t_3;
	elseif (t_4 <= 2e-7)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_2)) / (1.0 + x);
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+14], t$95$3, If[LessEqual[t$95$4, 2e-7], t$95$1, If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 84.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if -5e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 1.9999999999999999e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{t \cdot z - x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{t \cdot z - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7} \lor \neg \left(t\_1 \leq 100000000000\right):\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* y z) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -1e-24)
     (/ y t)
     (if (<= t_1 5e-157)
       (* (- 1.0 x) x)
       (if (or (<= t_1 2e-7) (not (<= t_1 100000000000.0))) (/ y t) 1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -1e-24) {
		tmp = y / t;
	} else if (t_1 <= 5e-157) {
		tmp = (1.0 - x) * x;
	} else if ((t_1 <= 2e-7) || !(t_1 <= 100000000000.0)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_1 <= (-1d-24)) then
        tmp = y / t
    else if (t_1 <= 5d-157) then
        tmp = (1.0d0 - x) * x
    else if ((t_1 <= 2d-7) .or. (.not. (t_1 <= 100000000000.0d0))) then
        tmp = y / t
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -1e-24) {
		tmp = y / t;
	} else if (t_1 <= 5e-157) {
		tmp = (1.0 - x) * x;
	} else if ((t_1 <= 2e-7) || !(t_1 <= 100000000000.0)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_1 <= -1e-24:
		tmp = y / t
	elif t_1 <= 5e-157:
		tmp = (1.0 - x) * x
	elif (t_1 <= 2e-7) or not (t_1 <= 100000000000.0):
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -1e-24)
		tmp = Float64(y / t);
	elseif (t_1 <= 5e-157)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif ((t_1 <= 2e-7) || !(t_1 <= 100000000000.0))
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -1e-24)
		tmp = y / t;
	elseif (t_1 <= 5e-157)
		tmp = (1.0 - x) * x;
	elseif ((t_1 <= 2e-7) || ~((t_1 <= 100000000000.0)))
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-24], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 5e-157], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[Or[LessEqual[t$95$1, 2e-7], N[Not[LessEqual[t$95$1, 100000000000.0]], $MachinePrecision]], N[(y / t), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.99999999999999924e-25 or 5.0000000000000002e-157 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7 or 1e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 51.7

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

   5. Applied rewrites 51.7%

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

   if -9.99999999999999924e-25 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000002e-157

   1. Initial program 96.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

      3. lower-+.f64 77.9

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   5. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.9%

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

   if 1.9999999999999999e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e11

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-7} \lor \neg \left(\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 100000000000\right):\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{\frac{y \cdot z - x}{t\_1} + x}{1 + x}\\ \mathbf{if}\;t\_2 \leq 0.9999999990577603:\\ \;\;\;\;\frac{\frac{y \cdot z}{t\_1} + x}{1 + x}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{1 + x}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t\_1} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ (/ (- (* y z) x) t_1) x) (+ 1.0 x))))
   (if (<= t_2 0.9999999990577603)
     (/ (+ (/ (* y z) t_1) x) (+ 1.0 x))
     (if (<= t_2 2.0)
       (/ (- x (/ x t_1)) (+ 1.0 x))
       (if (<= t_2 INFINITY)
         (/ (* (/ z t_1) y) (+ 1.0 x))
         (/ (+ (/ y t) x) (+ 1.0 x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = ((((y * z) - x) / t_1) + x) / (1.0 + x);
	double tmp;
	if (t_2 <= 0.9999999990577603) {
		tmp = (((y * z) / t_1) + x) / (1.0 + x);
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_1)) / (1.0 + x);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((z / t_1) * y) / (1.0 + x);
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = ((((y * z) - x) / t_1) + x) / (1.0 + x);
	double tmp;
	if (t_2 <= 0.9999999990577603) {
		tmp = (((y * z) / t_1) + x) / (1.0 + x);
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_1)) / (1.0 + x);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = ((z / t_1) * y) / (1.0 + x);
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = ((((y * z) - x) / t_1) + x) / (1.0 + x)
	tmp = 0
	if t_2 <= 0.9999999990577603:
		tmp = (((y * z) / t_1) + x) / (1.0 + x)
	elif t_2 <= 2.0:
		tmp = (x - (x / t_1)) / (1.0 + x)
	elif t_2 <= math.inf:
		tmp = ((z / t_1) * y) / (1.0 + x)
	else:
		tmp = ((y / t) + x) / (1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / t_1) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= 0.9999999990577603)
		tmp = Float64(Float64(Float64(Float64(y * z) / t_1) + x) / Float64(1.0 + x));
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(1.0 + x));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(z / t_1) * y) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = ((((y * z) - x) / t_1) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= 0.9999999990577603)
		tmp = (((y * z) / t_1) + x) / (1.0 + x);
	elseif (t_2 <= 2.0)
		tmp = (x - (x / t_1)) / (1.0 + x);
	elseif (t_2 <= Inf)
		tmp = ((z / t_1) * y) / (1.0 + x);
	else
		tmp = ((y / t) + x) / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.9999999990577603], N[(N[(N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(z / t$95$1), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999999999057760269

   1. Initial program 93.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if 0.999999999057760269 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 82.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 92.0

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

   5. Applied rewrites 92.0%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 0.9999999990577603:\\ \;\;\;\;\frac{\frac{y \cdot z}{t \cdot z - x} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t \cdot z - x} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;1 - \frac{z}{\left(1 + x\right) \cdot x} \cdot y\\ \mathbf{elif}\;t\_1 \leq 0.99999:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 100000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* y z) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -5e+14)
     (- 1.0 (* (/ z (* (+ 1.0 x) x)) y))
     (if (<= t_1 0.99999)
       (/ x (+ 1.0 x))
       (if (<= t_1 100000000000.0) 1.0 (/ (/ y t) (+ 1.0 x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e+14) {
		tmp = 1.0 - ((z / ((1.0 + x) * x)) * y);
	} else if (t_1 <= 0.99999) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 100000000000.0) {
		tmp = 1.0;
	} else {
		tmp = (y / t) / (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_1 <= (-5d+14)) then
        tmp = 1.0d0 - ((z / ((1.0d0 + x) * x)) * y)
    else if (t_1 <= 0.99999d0) then
        tmp = x / (1.0d0 + x)
    else if (t_1 <= 100000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = (y / t) / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e+14) {
		tmp = 1.0 - ((z / ((1.0 + x) * x)) * y);
	} else if (t_1 <= 0.99999) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 100000000000.0) {
		tmp = 1.0;
	} else {
		tmp = (y / t) / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_1 <= -5e+14:
		tmp = 1.0 - ((z / ((1.0 + x) * x)) * y)
	elif t_1 <= 0.99999:
		tmp = x / (1.0 + x)
	elif t_1 <= 100000000000.0:
		tmp = 1.0
	else:
		tmp = (y / t) / (1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e+14)
		tmp = Float64(1.0 - Float64(Float64(z / Float64(Float64(1.0 + x) * x)) * y));
	elseif (t_1 <= 0.99999)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_1 <= 100000000000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y / t) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -5e+14)
		tmp = 1.0 - ((z / ((1.0 + x) * x)) * y);
	elseif (t_1 <= 0.99999)
		tmp = x / (1.0 + x);
	elseif (t_1 <= 100000000000.0)
		tmp = 1.0;
	else
		tmp = (y / t) / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+14], N[(1.0 - N[(N[(z / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.99999], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 100000000000.0], 1.0, N[(N[(y / t), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e14

   1. Initial program 87.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 7.7%

      \[\leadsto \color{blue}{1} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-/r* N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 54.2

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

   8. Applied rewrites 54.2%

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

   if -5e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999990000000000046

   1. Initial program 97.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

      3. lower-+.f64 59.8

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   5. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if 0.999990000000000046 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e11

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{1} \]

   if 1e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 57.7

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

   5. Applied rewrites 57.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;1 - \frac{z}{\left(1 + x\right) \cdot x} \cdot y\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 0.99999:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 100000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t}}{1 + x}\\ t_2 := \frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.99999:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_2 \leq 100000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ y t) (+ 1.0 x)))
        (t_2 (/ (+ (/ (- (* y z) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_2 -5e+14)
     t_1
     (if (<= t_2 0.99999)
       (/ x (+ 1.0 x))
       (if (<= t_2 100000000000.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / t) / (1.0 + x);
	double t_2 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_2 <= -5e+14) {
		tmp = t_1;
	} else if (t_2 <= 0.99999) {
		tmp = x / (1.0 + x);
	} else if (t_2 <= 100000000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y / t) / (1.0d0 + x)
    t_2 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_2 <= (-5d+14)) then
        tmp = t_1
    else if (t_2 <= 0.99999d0) then
        tmp = x / (1.0d0 + x)
    else if (t_2 <= 100000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / t) / (1.0 + x);
	double t_2 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_2 <= -5e+14) {
		tmp = t_1;
	} else if (t_2 <= 0.99999) {
		tmp = x / (1.0 + x);
	} else if (t_2 <= 100000000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / t) / (1.0 + x)
	t_2 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_2 <= -5e+14:
		tmp = t_1
	elif t_2 <= 0.99999:
		tmp = x / (1.0 + x)
	elif t_2 <= 100000000000.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) / Float64(1.0 + x))
	t_2 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= -5e+14)
		tmp = t_1;
	elseif (t_2 <= 0.99999)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_2 <= 100000000000.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / t) / (1.0 + x);
	t_2 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= -5e+14)
		tmp = t_1;
	elseif (t_2 <= 0.99999)
		tmp = x / (1.0 + x);
	elseif (t_2 <= 100000000000.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+14], t$95$1, If[LessEqual[t$95$2, 0.99999], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 100000000000.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e14 or 1e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 53.4

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

   5. Applied rewrites 53.4%

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

   if -5e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999990000000000046

   1. Initial program 97.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

      3. lower-+.f64 59.8

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   5. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if 0.999990000000000046 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e11

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{t}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 0.99999:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 100000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.99999:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 100000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* y z) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -5e+14)
     (/ y t)
     (if (<= t_1 0.99999)
       (/ x (+ 1.0 x))
       (if (<= t_1 100000000000.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e+14) {
		tmp = y / t;
	} else if (t_1 <= 0.99999) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 100000000000.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_1 <= (-5d+14)) then
        tmp = y / t
    else if (t_1 <= 0.99999d0) then
        tmp = x / (1.0d0 + x)
    else if (t_1 <= 100000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e+14) {
		tmp = y / t;
	} else if (t_1 <= 0.99999) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 100000000000.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_1 <= -5e+14:
		tmp = y / t
	elif t_1 <= 0.99999:
		tmp = x / (1.0 + x)
	elif t_1 <= 100000000000.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e+14)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.99999)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_1 <= 100000000000.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -5e+14)
		tmp = y / t;
	elseif (t_1 <= 0.99999)
		tmp = x / (1.0 + x);
	elseif (t_1 <= 100000000000.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+14], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.99999], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 100000000000.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e14 or 1e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -5e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999990000000000046

   1. Initial program 97.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

      3. lower-+.f64 59.8

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   5. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if 0.999990000000000046 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e11

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 0.99999:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 100000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{\frac{y \cdot z - x}{t\_1} + x}{1 + x}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-7} \lor \neg \left(t\_2 \leq 100000000000\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ (/ (- (* y z) x) t_1) x) (+ 1.0 x))))
   (if (or (<= t_2 2e-7) (not (<= t_2 100000000000.0)))
     (/ (+ (/ y t) x) (+ 1.0 x))
     (/ (- x (/ x t_1)) (+ 1.0 x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = ((((y * z) - x) / t_1) + x) / (1.0 + x);
	double tmp;
	if ((t_2 <= 2e-7) || !(t_2 <= 100000000000.0)) {
		tmp = ((y / t) + x) / (1.0 + x);
	} else {
		tmp = (x - (x / t_1)) / (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * z) - x
    t_2 = ((((y * z) - x) / t_1) + x) / (1.0d0 + x)
    if ((t_2 <= 2d-7) .or. (.not. (t_2 <= 100000000000.0d0))) then
        tmp = ((y / t) + x) / (1.0d0 + x)
    else
        tmp = (x - (x / t_1)) / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = ((((y * z) - x) / t_1) + x) / (1.0 + x);
	double tmp;
	if ((t_2 <= 2e-7) || !(t_2 <= 100000000000.0)) {
		tmp = ((y / t) + x) / (1.0 + x);
	} else {
		tmp = (x - (x / t_1)) / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = ((((y * z) - x) / t_1) + x) / (1.0 + x)
	tmp = 0
	if (t_2 <= 2e-7) or not (t_2 <= 100000000000.0):
		tmp = ((y / t) + x) / (1.0 + x)
	else:
		tmp = (x - (x / t_1)) / (1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / t_1) + x) / Float64(1.0 + x))
	tmp = 0.0
	if ((t_2 <= 2e-7) || !(t_2 <= 100000000000.0))
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = ((((y * z) - x) / t_1) + x) / (1.0 + x);
	tmp = 0.0;
	if ((t_2 <= 2e-7) || ~((t_2 <= 100000000000.0)))
		tmp = ((y / t) + x) / (1.0 + x);
	else
		tmp = (x - (x / t_1)) / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, 2e-7], N[Not[LessEqual[t$95$2, 100000000000.0]], $MachinePrecision]], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7 or 1e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 83.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 1.9999999999999999e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e11

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-7} \lor \neg \left(\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 100000000000\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq 0.99999 \lor \neg \left(t\_1 \leq 100000000000\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* y z) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (or (<= t_1 0.99999) (not (<= t_1 100000000000.0)))
     (/ (+ (/ y t) x) (+ 1.0 x))
     1.0)))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if ((t_1 <= 0.99999) || !(t_1 <= 100000000000.0)) {
		tmp = ((y / t) + x) / (1.0 + x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if ((t_1 <= 0.99999d0) .or. (.not. (t_1 <= 100000000000.0d0))) then
        tmp = ((y / t) + x) / (1.0d0 + x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if ((t_1 <= 0.99999) || !(t_1 <= 100000000000.0)) {
		tmp = ((y / t) + x) / (1.0 + x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if (t_1 <= 0.99999) or not (t_1 <= 100000000000.0):
		tmp = ((y / t) + x) / (1.0 + x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if ((t_1 <= 0.99999) || !(t_1 <= 100000000000.0))
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if ((t_1 <= 0.99999) || ~((t_1 <= 100000000000.0)))
		tmp = ((y / t) + x) / (1.0 + x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.99999], N[Not[LessEqual[t$95$1, 100000000000.0]], $MachinePrecision]], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999990000000000046 or 1e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 83.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if 0.999990000000000046 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e11

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 0.99999 \lor \neg \left(\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 100000000000\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+233}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* y z) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 2e+233) t_1 (/ (+ (/ y t) x) (+ 1.0 x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= 2e+233) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_1 <= 2d+233) then
        tmp = t_1
    else
        tmp = ((y / t) + x) / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= 2e+233) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_1 <= 2e+233:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= 2e+233)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= 2e+233)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+233], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999995e233

   1. Initial program 97.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   if 1.99999999999999995e233 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 30.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ (/ (- (* y z) x) (- (* t z) x)) x) (+ 1.0 x)) 2e-16)
   (* (- 1.0 x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 2e-16) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((((y * z) - x) / ((t * z) - x)) + x) / (1.0d0 + x)) <= 2d-16) then
        tmp = (1.0d0 - x) * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 2e-16) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 2e-16:
		tmp = (1.0 - x) * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x)) <= 2e-16)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((((((y * z) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 2e-16)
		tmp = (1.0 - x) * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 2e-16], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-16

   1. Initial program 93.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

      3. lower-+.f64 39.7

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   5. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.5%

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

   if 2e-16 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 91.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y \cdot z - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.6% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 92.3%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 56.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0);
}

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(y / Float64(t - Float64(x / z))) - Float64(x / Float64(Float64(t * z) - x)))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(y / N[(t - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024271 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1)))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))