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

Percentage Accurate: 88.6% → 97.0%

Time: 9.5s

Alternatives: 18

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: 88.6% 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: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - t \cdot z\\ t_2 := \frac{z}{x - -1}\\ t_3 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ t_4 := \frac{-1}{t\_1}\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t\_4, t\_2, \frac{\frac{x}{y}}{t\_1 \cdot \left(x - -1\right)} + \frac{x}{\left(x - -1\right) \cdot y}\right) \cdot y\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t\_4, t\_2, \frac{1}{y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (* t z)))
        (t_2 (/ z (- x -1.0)))
        (t_3 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)))
        (t_4 (/ -1.0 t_1)))
   (if (<= t_3 2e-6)
     (*
      (fma t_4 t_2 (+ (/ (/ x y) (* t_1 (- x -1.0))) (/ x (* (- x -1.0) y))))
      y)
     (if (<= t_3 INFINITY)
       (* (fma t_4 t_2 (/ 1.0 y)) y)
       (/ (+ (/ y t) x) (- x -1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (t * z);
	double t_2 = z / (x - -1.0);
	double t_3 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_4 = -1.0 / t_1;
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = fma(t_4, t_2, (((x / y) / (t_1 * (x - -1.0))) + (x / ((x - -1.0) * y)))) * y;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma(t_4, t_2, (1.0 / y)) * y;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(t * z))
	t_2 = Float64(z / Float64(x - -1.0))
	t_3 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	t_4 = Float64(-1.0 / t_1)
	tmp = 0.0
	if (t_3 <= 2e-6)
		tmp = Float64(fma(t_4, t_2, Float64(Float64(Float64(x / y) / Float64(t_1 * Float64(x - -1.0))) + Float64(x / Float64(Float64(x - -1.0) * y)))) * y);
	elseif (t_3 <= Inf)
		tmp = Float64(fma(t_4, t_2, Float64(1.0 / y)) * y);
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-6], N[(N[(t$95$4 * t$95$2 + N[(N[(N[(x / y), $MachinePrecision] / N[(t$95$1 * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(N[(x - -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$4 * t$95$2 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $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))) < 1.99999999999999991e-6

   1. Initial program 94.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 93.9

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

      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 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 98.4%

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

   if 1.99999999999999991e-6 < (/.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 93.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 93.9

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

      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 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 62.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.5%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x - t \cdot z}, \frac{z}{x - -1}, \frac{\frac{x}{y}}{\left(x - t \cdot z\right) \cdot \left(x - -1\right)} + \frac{x}{\left(x - -1\right) \cdot y}\right) \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x - t \cdot z}, \frac{z}{x - -1}, \frac{1}{y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ t_2 := x - t \cdot z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{t\_2 \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{\frac{-1 - x}{z} \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)))
        (t_2 (- x (* t z))))
   (if (<= t_1 -5e+31)
     (* (/ z (* t_2 (- -1.0 x))) y)
     (if (<= t_1 1e-23)
       (/ (- x (/ (- (/ x z) y) t)) 1.0)
       (if (<= t_1 2.0)
         (/ (- x (/ x (fma t z (- x)))) (- x -1.0))
         (if (<= t_1 INFINITY)
           (/ y (* (/ (- -1.0 x) z) t_2))
           (/ (+ (/ y t) x) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_2 = x - (t * z);
	double tmp;
	if (t_1 <= -5e+31) {
		tmp = (z / (t_2 * (-1.0 - x))) * y;
	} else if (t_1 <= 1e-23) {
		tmp = (x - (((x / z) - y) / t)) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / fma(t, z, -x))) / (x - -1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / (((-1.0 - x) / z) * t_2);
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	t_2 = Float64(x - Float64(t * z))
	tmp = 0.0
	if (t_1 <= -5e+31)
		tmp = Float64(Float64(z / Float64(t_2 * Float64(-1.0 - x))) * y);
	elseif (t_1 <= 1e-23)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(x - -1.0));
	elseif (t_1 <= Inf)
		tmp = Float64(y / Float64(Float64(Float64(-1.0 - x) / z) * t_2));
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+31], N[(N[(z / N[(t$95$2 * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e-23], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(y / N[(N[(N[(-1.0 - x), $MachinePrecision] / z), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000027e31

   1. Initial program 85.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 96.0%

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

   if -5.00000000000000027e31 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999996e-24

   1. Initial program 98.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg 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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   if 9.9999999999999996e-24 < (/.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. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

   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 73.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 80.4

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 88.3%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\frac{y}{t} + x}}{x + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10^{-23}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{y}{\frac{-1 - x}{z} \cdot \left(x - t \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x - -1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t\_2} \cdot y}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)))
        (t_2 (fma t z (- x))))
   (if (<= t_1 -5e+31)
     (* (/ z (* (- x (* t z)) (- -1.0 x))) y)
     (if (<= t_1 1e-23)
       (/ (- x (/ (- (/ x z) y) t)) 1.0)
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (- x -1.0))
         (if (<= t_1 INFINITY)
           (/ (* (/ z t_2) y) (- x -1.0))
           (/ (+ (/ y t) x) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_2 = fma(t, z, -x);
	double tmp;
	if (t_1 <= -5e+31) {
		tmp = (z / ((x - (t * z)) * (-1.0 - x))) * y;
	} else if (t_1 <= 1e-23) {
		tmp = (x - (((x / z) - y) / t)) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((z / t_2) * y) / (x - -1.0);
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	t_2 = fma(t, z, Float64(-x))
	tmp = 0.0
	if (t_1 <= -5e+31)
		tmp = Float64(Float64(z / Float64(Float64(x - Float64(t * z)) * Float64(-1.0 - x))) * y);
	elseif (t_1 <= 1e-23)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x - -1.0));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(z / t_2) * y) / Float64(x - -1.0));
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+31], N[(N[(z / N[(N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e-23], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000027e31

   1. Initial program 85.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 96.0%

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

   if -5.00000000000000027e31 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999996e-24

   1. Initial program 98.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg 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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   if 9.9999999999999996e-24 < (/.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. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

   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 73.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 85.7

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

   5. Applied rewrites 85.7%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\frac{y}{t} + x}}{x + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10^{-23}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} + x\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ t_3 := \mathsf{fma}\left(t, z, -x\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\frac{t\_1}{1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_3}}{x - -1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t\_3} \cdot y}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y t) x))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)))
        (t_3 (fma t z (- x))))
   (if (<= t_2 -5e+31)
     (* (/ z (* (- x (* t z)) (- -1.0 x))) y)
     (if (<= t_2 2e-62)
       (/ t_1 1.0)
       (if (<= t_2 2.0)
         (/ (- x (/ x t_3)) (- x -1.0))
         (if (<= t_2 INFINITY)
           (/ (* (/ z t_3) y) (- x -1.0))
           (/ t_1 (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / t) + x;
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_3 = fma(t, z, -x);
	double tmp;
	if (t_2 <= -5e+31) {
		tmp = (z / ((x - (t * z)) * (-1.0 - x))) * y;
	} else if (t_2 <= 2e-62) {
		tmp = t_1 / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_3)) / (x - -1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((z / t_3) * y) / (x - -1.0);
	} else {
		tmp = t_1 / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) + x)
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	t_3 = fma(t, z, Float64(-x))
	tmp = 0.0
	if (t_2 <= -5e+31)
		tmp = Float64(Float64(z / Float64(Float64(x - Float64(t * z)) * Float64(-1.0 - x))) * y);
	elseif (t_2 <= 2e-62)
		tmp = Float64(t_1 / 1.0);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_3)) / Float64(x - -1.0));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(z / t_3) * y) / Float64(x - -1.0));
	else
		tmp = Float64(t_1 / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * z + (-x)), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+31], N[(N[(z / N[(N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, 2e-62], N[(t$95$1 / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(z / t$95$3), $MachinePrecision] * y), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000027e31

   1. Initial program 85.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 96.0%

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

   if -5.00000000000000027e31 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-62

   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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.6%

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

   if 2.0000000000000001e-62 < (/.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. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 98.4

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

   5. Applied rewrites 98.4%

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

   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 73.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 85.7

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

   5. Applied rewrites 85.7%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\frac{y}{t} + x}}{x + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ t_3 := \frac{y}{t} + x\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\frac{t\_3}{1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (* (- x (* t z)) (- -1.0 x))) y))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)))
        (t_3 (+ (/ y t) x)))
   (if (<= t_2 -5e+31)
     t_1
     (if (<= t_2 2e-62)
       (/ t_3 1.0)
       (if (<= t_2 2.0)
         (/ (- x (/ x (fma t z (- x)))) (- x -1.0))
         (if (<= t_2 INFINITY) t_1 (/ t_3 (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / ((x - (t * z)) * (-1.0 - x))) * y;
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_3 = (y / t) + x;
	double tmp;
	if (t_2 <= -5e+31) {
		tmp = t_1;
	} else if (t_2 <= 2e-62) {
		tmp = t_3 / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / fma(t, z, -x))) / (x - -1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3 / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / Float64(Float64(x - Float64(t * z)) * Float64(-1.0 - x))) * y)
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	t_3 = Float64(Float64(y / t) + x)
	tmp = 0.0
	if (t_2 <= -5e+31)
		tmp = t_1;
	elseif (t_2 <= 2e-62)
		tmp = Float64(t_3 / 1.0);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(x - -1.0));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(t_3 / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+31], t$95$1, If[LessEqual[t$95$2, 2e-62], N[(t$95$3 / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(t$95$3 / N[(x - -1.0), $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))) < -5.00000000000000027e31 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 78.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 89.8%

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

   if -5.00000000000000027e31 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-62

   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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.6%

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

   if 2.0000000000000001e-62 < (/.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. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 98.4

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

   5. Applied rewrites 98.4%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ t_3 := \frac{y}{t} + x\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-23}:\\ \;\;\;\;\frac{t\_3}{1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (* (- x (* t z)) (- -1.0 x))) y))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)))
        (t_3 (+ (/ y t) x)))
   (if (<= t_2 -5e+31)
     t_1
     (if (<= t_2 1e-23)
       (/ t_3 1.0)
       (if (<= t_2 2.0) 1.0 (if (<= t_2 INFINITY) t_1 (/ t_3 (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / ((x - (t * z)) * (-1.0 - x))) * y;
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_3 = (y / t) + x;
	double tmp;
	if (t_2 <= -5e+31) {
		tmp = t_1;
	} else if (t_2 <= 1e-23) {
		tmp = t_3 / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3 / (x - -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / ((x - (t * z)) * (-1.0 - x))) * y;
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_3 = (y / t) + x;
	double tmp;
	if (t_2 <= -5e+31) {
		tmp = t_1;
	} else if (t_2 <= 1e-23) {
		tmp = t_3 / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_3 / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / ((x - (t * z)) * (-1.0 - x))) * y
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	t_3 = (y / t) + x
	tmp = 0
	if t_2 <= -5e+31:
		tmp = t_1
	elif t_2 <= 1e-23:
		tmp = t_3 / 1.0
	elif t_2 <= 2.0:
		tmp = 1.0
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = t_3 / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / Float64(Float64(x - Float64(t * z)) * Float64(-1.0 - x))) * y)
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	t_3 = Float64(Float64(y / t) + x)
	tmp = 0.0
	if (t_2 <= -5e+31)
		tmp = t_1;
	elseif (t_2 <= 1e-23)
		tmp = Float64(t_3 / 1.0);
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(t_3 / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / ((x - (t * z)) * (-1.0 - x))) * y;
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	t_3 = (y / t) + x;
	tmp = 0.0;
	if (t_2 <= -5e+31)
		tmp = t_1;
	elseif (t_2 <= 1e-23)
		tmp = t_3 / 1.0;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3 / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+31], t$95$1, If[LessEqual[t$95$2, 1e-23], N[(t$95$3 / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, Infinity], t$95$1, N[(t$95$3 / N[(x - -1.0), $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))) < -5.00000000000000027e31 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 78.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 89.8%

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

   if -5.00000000000000027e31 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999996e-24

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 87.3%

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

   if 9.9999999999999996e-24 < (/.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 x around inf

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

   5. Applied rewrites 97.5%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{z}{\left(x - t \cdot z\right) \cdot \left(-1 - x\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} + x\\ t_2 := t \cdot z - x\\ t_3 := \frac{x - \frac{x - z \cdot y}{t\_2}}{x - -1}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{z \cdot y}{1 \cdot t\_2}\\ \mathbf{elif}\;t\_3 \leq 10^{-23}:\\ \;\;\;\;\frac{t\_1}{1}\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot y}{\left(-x\right) \cdot \left(x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y t) x))
        (t_2 (- (* t z) x))
        (t_3 (/ (- x (/ (- x (* z y)) t_2)) (- x -1.0))))
   (if (<= t_3 -2e-35)
     (/ (* z y) (* 1.0 t_2))
     (if (<= t_3 1e-23)
       (/ t_1 1.0)
       (if (<= t_3 10.0)
         1.0
         (if (<= t_3 5e+97)
           (/ (* z y) (* (- x) (- x -1.0)))
           (/ t_1 (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / t) + x;
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double tmp;
	if (t_3 <= -2e-35) {
		tmp = (z * y) / (1.0 * t_2);
	} else if (t_3 <= 1e-23) {
		tmp = t_1 / 1.0;
	} else if (t_3 <= 10.0) {
		tmp = 1.0;
	} else if (t_3 <= 5e+97) {
		tmp = (z * y) / (-x * (x - -1.0));
	} else {
		tmp = t_1 / (x - -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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (y / t) + x
    t_2 = (t * z) - x
    t_3 = (x - ((x - (z * y)) / t_2)) / (x - (-1.0d0))
    if (t_3 <= (-2d-35)) then
        tmp = (z * y) / (1.0d0 * t_2)
    else if (t_3 <= 1d-23) then
        tmp = t_1 / 1.0d0
    else if (t_3 <= 10.0d0) then
        tmp = 1.0d0
    else if (t_3 <= 5d+97) then
        tmp = (z * y) / (-x * (x - (-1.0d0)))
    else
        tmp = t_1 / (x - (-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 / t) + x;
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double tmp;
	if (t_3 <= -2e-35) {
		tmp = (z * y) / (1.0 * t_2);
	} else if (t_3 <= 1e-23) {
		tmp = t_1 / 1.0;
	} else if (t_3 <= 10.0) {
		tmp = 1.0;
	} else if (t_3 <= 5e+97) {
		tmp = (z * y) / (-x * (x - -1.0));
	} else {
		tmp = t_1 / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / t) + x
	t_2 = (t * z) - x
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0)
	tmp = 0
	if t_3 <= -2e-35:
		tmp = (z * y) / (1.0 * t_2)
	elif t_3 <= 1e-23:
		tmp = t_1 / 1.0
	elif t_3 <= 10.0:
		tmp = 1.0
	elif t_3 <= 5e+97:
		tmp = (z * y) / (-x * (x - -1.0))
	else:
		tmp = t_1 / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) + x)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / t_2)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_3 <= -2e-35)
		tmp = Float64(Float64(z * y) / Float64(1.0 * t_2));
	elseif (t_3 <= 1e-23)
		tmp = Float64(t_1 / 1.0);
	elseif (t_3 <= 10.0)
		tmp = 1.0;
	elseif (t_3 <= 5e+97)
		tmp = Float64(Float64(z * y) / Float64(Float64(-x) * Float64(x - -1.0)));
	else
		tmp = Float64(t_1 / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / t) + x;
	t_2 = (t * z) - x;
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	tmp = 0.0;
	if (t_3 <= -2e-35)
		tmp = (z * y) / (1.0 * t_2);
	elseif (t_3 <= 1e-23)
		tmp = t_1 / 1.0;
	elseif (t_3 <= 10.0)
		tmp = 1.0;
	elseif (t_3 <= 5e+97)
		tmp = (z * y) / (-x * (x - -1.0));
	else
		tmp = t_1 / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-35], N[(N[(z * y), $MachinePrecision] / N[(1.0 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-23], N[(t$95$1 / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 10.0], 1.0, If[LessEqual[t$95$3, 5e+97], N[(N[(z * y), $MachinePrecision] / N[((-x) * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000002e-35

   1. Initial program 87.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 79.7

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

   5. Applied rewrites 79.7%

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 75.6%

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

   if -2.00000000000000002e-35 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999996e-24

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.1%

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

   if 9.9999999999999996e-24 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 10

   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 x around inf

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

   5. Applied rewrites 96.9%

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

   if 10 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999999e97

   1. Initial program 99.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.8

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.7%

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

   if 4.99999999999999999e97 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 39.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

      \[\leadsto \frac{\color{blue}{\frac{y}{t} + x}}{x + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{z \cdot y}{1 \cdot \left(t \cdot z - x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot y}{\left(-x\right) \cdot \left(x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{\left(-x\right) \cdot \left(x - -1\right)}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;t\_2 \leq 10:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z y) (* (- x) (- x -1.0))))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_2 -4e+173)
     t_1
     (if (<= t_2 1e-23)
       (/ (+ (/ y t) x) 1.0)
       (if (<= t_2 10.0)
         1.0
         (if (<= t_2 5e+97) t_1 (/ y (* (- x -1.0) t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / (-x * (x - -1.0));
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= -4e+173) {
		tmp = t_1;
	} else if (t_2 <= 1e-23) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_2 <= 10.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+97) {
		tmp = t_1;
	} else {
		tmp = y / ((x - -1.0) * 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) :: t_2
    real(8) :: tmp
    t_1 = (z * y) / (-x * (x - (-1.0d0)))
    t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_2 <= (-4d+173)) then
        tmp = t_1
    else if (t_2 <= 1d-23) then
        tmp = ((y / t) + x) / 1.0d0
    else if (t_2 <= 10.0d0) then
        tmp = 1.0d0
    else if (t_2 <= 5d+97) then
        tmp = t_1
    else
        tmp = y / ((x - (-1.0d0)) * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / (-x * (x - -1.0));
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= -4e+173) {
		tmp = t_1;
	} else if (t_2 <= 1e-23) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_2 <= 10.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+97) {
		tmp = t_1;
	} else {
		tmp = y / ((x - -1.0) * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * y) / (-x * (x - -1.0))
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_2 <= -4e+173:
		tmp = t_1
	elif t_2 <= 1e-23:
		tmp = ((y / t) + x) / 1.0
	elif t_2 <= 10.0:
		tmp = 1.0
	elif t_2 <= 5e+97:
		tmp = t_1
	else:
		tmp = y / ((x - -1.0) * t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) / Float64(Float64(-x) * Float64(x - -1.0)))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= -4e+173)
		tmp = t_1;
	elseif (t_2 <= 1e-23)
		tmp = Float64(Float64(Float64(y / t) + x) / 1.0);
	elseif (t_2 <= 10.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+97)
		tmp = t_1;
	else
		tmp = Float64(y / Float64(Float64(x - -1.0) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * y) / (-x * (x - -1.0));
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_2 <= -4e+173)
		tmp = t_1;
	elseif (t_2 <= 1e-23)
		tmp = ((y / t) + x) / 1.0;
	elseif (t_2 <= 10.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+97)
		tmp = t_1;
	else
		tmp = y / ((x - -1.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / N[((-x) * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+173], t$95$1, If[LessEqual[t$95$2, 1e-23], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 10.0], 1.0, If[LessEqual[t$95$2, 5e+97], t$95$1, N[(y / N[(N[(x - -1.0), $MachinePrecision] * t), $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))) < -4.0000000000000001e173 or 10 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999999e97

   1. Initial program 85.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 70.6%

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

   if -4.0000000000000001e173 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999996e-24

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.6%

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

   if 9.9999999999999996e-24 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 10

   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 x around inf

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

   5. Applied rewrites 96.9%

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

   if 4.99999999999999999e97 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 39.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -4 \cdot 10^{+173}:\\ \;\;\;\;\frac{z \cdot y}{\left(-x\right) \cdot \left(x - -1\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot y}{\left(-x\right) \cdot \left(x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x - -1}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_2 \leq 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot y}{\left(-x\right) \cdot \left(x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (- x -1.0)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_2 1e-23)
     t_1
     (if (<= t_2 10.0)
       1.0
       (if (<= t_2 5e+97) (/ (* z y) (* (- x) (- x -1.0))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= 1e-23) {
		tmp = t_1;
	} else if (t_2 <= 10.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+97) {
		tmp = (z * y) / (-x * (x - -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) + x) / (x - (-1.0d0))
    t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_2 <= 1d-23) then
        tmp = t_1
    else if (t_2 <= 10.0d0) then
        tmp = 1.0d0
    else if (t_2 <= 5d+97) then
        tmp = (z * y) / (-x * (x - (-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) + x) / (x - -1.0);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= 1e-23) {
		tmp = t_1;
	} else if (t_2 <= 10.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+97) {
		tmp = (z * y) / (-x * (x - -1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x - -1.0)
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_2 <= 1e-23:
		tmp = t_1
	elif t_2 <= 10.0:
		tmp = 1.0
	elif t_2 <= 5e+97:
		tmp = (z * y) / (-x * (x - -1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= 1e-23)
		tmp = t_1;
	elseif (t_2 <= 10.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+97)
		tmp = Float64(Float64(z * y) / Float64(Float64(-x) * Float64(x - -1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x - -1.0);
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_2 <= 1e-23)
		tmp = t_1;
	elseif (t_2 <= 10.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+97)
		tmp = (z * y) / (-x * (x - -1.0));
	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[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-23], t$95$1, If[LessEqual[t$95$2, 10.0], 1.0, If[LessEqual[t$95$2, 5e+97], N[(N[(z * y), $MachinePrecision] / N[((-x) * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < 9.9999999999999996e-24 or 4.99999999999999999e97 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 77.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if 9.9999999999999996e-24 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 10

   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 x around inf

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

   5. Applied rewrites 96.9%

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

   if 10 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999999e97

   1. Initial program 99.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.8

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot y}{\left(-x\right) \cdot \left(x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - \frac{z}{x \cdot x} \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 (- INFINITY))
     (- 1.0 (* (/ z (* x x)) y))
     (if (<= t_1 1e-23)
       (/ (+ (/ y t) x) 1.0)
       (if (<= t_1 2.0) 1.0 (/ y (* (- x -1.0) t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 - ((z / (x * x)) * y);
	} else if (t_1 <= 1e-23) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / ((x - -1.0) * t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - ((z / (x * x)) * y);
	} else if (t_1 <= 1e-23) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / ((x - -1.0) * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 1.0 - ((z / (x * x)) * y)
	elif t_1 <= 1e-23:
		tmp = ((y / t) + x) / 1.0
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / ((x - -1.0) * t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 - Float64(Float64(z / Float64(x * x)) * y));
	elseif (t_1 <= 1e-23)
		tmp = Float64(Float64(Float64(y / t) + x) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / Float64(Float64(x - -1.0) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 1.0 - ((z / (x * x)) * y);
	elseif (t_1 <= 1e-23)
		tmp = ((y / t) + x) / 1.0;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / ((x - -1.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(1.0 - N[(N[(z / N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-23], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / N[(N[(x - -1.0), $MachinePrecision] * t), $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))) < -inf.0

   1. Initial program 57.5%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{y \cdot z - t \cdot z}{{x}^{2}}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{y \cdot z - t \cdot z}{{x}^{2}}} \]

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 - \frac{y \cdot z}{\color{blue}{{x}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999996e-24

   1. Initial program 98.3%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.6%

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

   if 9.9999999999999996e-24 < (/.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 x around inf

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

   5. Applied rewrites 97.5%

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

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

   1. Initial program 54.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 51.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -\infty:\\ \;\;\;\;1 - \frac{z}{x \cdot x} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(x - -1\right) \cdot t}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.9999999999602612:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (- x -1.0) t)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_2 -5e-18)
     t_1
     (if (<= t_2 0.9999999999602612)
       (/ x (- x -1.0))
       (if (<= t_2 2.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / ((x - -1.0) * t);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= -5e-18) {
		tmp = t_1;
	} else if (t_2 <= 0.9999999999602612) {
		tmp = x / (x - -1.0);
	} else if (t_2 <= 2.0) {
		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 / ((x - (-1.0d0)) * t)
    t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_2 <= (-5d-18)) then
        tmp = t_1
    else if (t_2 <= 0.9999999999602612d0) then
        tmp = x / (x - (-1.0d0))
    else if (t_2 <= 2.0d0) 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 / ((x - -1.0) * t);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= -5e-18) {
		tmp = t_1;
	} else if (t_2 <= 0.9999999999602612) {
		tmp = x / (x - -1.0);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / ((x - -1.0) * t)
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_2 <= -5e-18:
		tmp = t_1
	elif t_2 <= 0.9999999999602612:
		tmp = x / (x - -1.0)
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(x - -1.0) * t))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= -5e-18)
		tmp = t_1;
	elseif (t_2 <= 0.9999999999602612)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / ((x - -1.0) * t);
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_2 <= -5e-18)
		tmp = t_1;
	elseif (t_2 <= 0.9999999999602612)
		tmp = x / (x - -1.0);
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(x - -1.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-18], t$95$1, If[LessEqual[t$95$2, 0.9999999999602612], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 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))) < -5.00000000000000036e-18 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.5%

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

   if -5.00000000000000036e-18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999999999960261232

   1. Initial program 98.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. lower-+.f64 55.8

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

   5. Applied rewrites 55.8%

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

   if 0.999999999960261232 < (/.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 x around inf

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

   5. Applied rewrites 98.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.9999999999602612:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999602612:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 -5e-18)
     (/ y t)
     (if (<= t_1 0.9999999999602612)
       (/ x (- x -1.0))
       (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -5e-18) {
		tmp = y / t;
	} else if (t_1 <= 0.9999999999602612) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 2.0) {
		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 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= (-5d-18)) then
        tmp = y / t
    else if (t_1 <= 0.9999999999602612d0) then
        tmp = x / (x - (-1.0d0))
    else if (t_1 <= 2.0d0) 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 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -5e-18) {
		tmp = y / t;
	} else if (t_1 <= 0.9999999999602612) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= -5e-18:
		tmp = y / t
	elif t_1 <= 0.9999999999602612:
		tmp = x / (x - -1.0)
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= -5e-18)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.9999999999602612)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= -5e-18)
		tmp = y / t;
	elseif (t_1 <= 0.9999999999602612)
		tmp = x / (x - -1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-18], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999602612], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 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))) < -5.00000000000000036e-18 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 46.3

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

   5. Applied rewrites 46.3%

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

   if -5.00000000000000036e-18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999999999960261232

   1. Initial program 98.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. lower-+.f64 55.8

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

   5. Applied rewrites 55.8%

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

   if 0.999999999960261232 < (/.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 x around inf

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

   5. Applied rewrites 98.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.9999999999602612:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 -5e-18)
     (/ y t)
     (if (<= t_1 2e-30) (* (- 1.0 x) x) (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -5e-18) {
		tmp = y / t;
	} else if (t_1 <= 2e-30) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 2.0) {
		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 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= (-5d-18)) then
        tmp = y / t
    else if (t_1 <= 2d-30) then
        tmp = (1.0d0 - x) * x
    else if (t_1 <= 2.0d0) 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 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -5e-18) {
		tmp = y / t;
	} else if (t_1 <= 2e-30) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= -5e-18:
		tmp = y / t
	elif t_1 <= 2e-30:
		tmp = (1.0 - x) * x
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= -5e-18)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-30)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= -5e-18)
		tmp = y / t;
	elseif (t_1 <= 2e-30)
		tmp = (1.0 - x) * x;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-18], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-30], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 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))) < -5.00000000000000036e-18 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 46.3

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

   5. Applied rewrites 46.3%

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

   if -5.00000000000000036e-18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-30

   1. Initial program 97.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. lower-+.f64 58.1

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

   5. Applied rewrites 58.1%

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

   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 58.1%

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

   if 2e-30 < (/.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 x around inf

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

   5. Applied rewrites 96.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ t_2 := x - t \cdot z\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t\_2}{z \cdot y - x}} + x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t\_2}, \frac{z}{x - -1}, \frac{1}{y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)))
        (t_2 (- x (* t z))))
   (if (<= t_1 2e+85)
     (/ (+ (/ -1.0 (/ t_2 (- (* z y) x))) x) (- x -1.0))
     (if (<= t_1 INFINITY)
       (* (fma (/ -1.0 t_2) (/ z (- x -1.0)) (/ 1.0 y)) y)
       (/ (+ (/ y t) x) (- x -1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_2 = x - (t * z);
	double tmp;
	if (t_1 <= 2e+85) {
		tmp = ((-1.0 / (t_2 / ((z * y) - x))) + x) / (x - -1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((-1.0 / t_2), (z / (x - -1.0)), (1.0 / y)) * y;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	t_2 = Float64(x - Float64(t * z))
	tmp = 0.0
	if (t_1 <= 2e+85)
		tmp = Float64(Float64(Float64(-1.0 / Float64(t_2 / Float64(Float64(z * y) - x))) + x) / Float64(x - -1.0));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(-1.0 / t_2), Float64(z / Float64(x - -1.0)), Float64(1.0 / y)) * y);
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+85], N[(N[(N[(-1.0 / N[(t$95$2 / N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(-1.0 / t$95$2), $MachinePrecision] * N[(z / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $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))) < 2e85

   1. Initial program 97.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 97.6

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

      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 97.6

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

   4. Applied rewrites 97.6%

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

   if 2e85 < (/.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 61.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 61.3

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

      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 61.3

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

   4. Applied rewrites 61.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.4%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{-1}{\frac{x - t \cdot z}{z \cdot y - x}} + x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x - t \cdot z}, \frac{z}{x - -1}, \frac{1}{y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x - t \cdot z}, \frac{z}{x - -1}, \frac{1}{y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 5e+111)
     t_1
     (if (<= t_1 INFINITY)
       (* (fma (/ -1.0 (- x (* t z))) (/ z (- x -1.0)) (/ 1.0 y)) y)
       (/ (+ (/ y t) x) (- x -1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 5e+111) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((-1.0 / (x - (t * z))), (z / (x - -1.0)), (1.0 / y)) * y;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 5e+111)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(-1.0 / Float64(x - Float64(t * z))), Float64(z / Float64(x - -1.0)), Float64(1.0 / y)) * y);
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+111], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(N[(-1.0 / N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $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))) < 4.9999999999999997e111

   1. Initial program 97.7%

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

   if 4.9999999999999997e111 < (/.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 56.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 56.2

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

      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 56.2

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

   4. Applied rewrites 56.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.5%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x - t \cdot z}, \frac{z}{x - -1}, \frac{1}{y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+194}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 2e+194) t_1 (/ (+ (/ y t) x) (- x -1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 2e+194) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -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 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= 2d+194) then
        tmp = t_1
    else
        tmp = ((y / t) + x) / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 2e+194) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= 2e+194:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 2e+194)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= 2e+194)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+194], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $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))) < 1.99999999999999989e194

   1. Initial program 97.7%

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

   if 1.99999999999999989e194 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 25.6%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 79.8

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

   5. Applied rewrites 79.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)) 2e-30)
   (* (- 1.0 x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)) <= 2e-30) {
		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 (((x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))) <= 2d-30) 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 (((x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)) <= 2e-30) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)) <= 2e-30:
		tmp = (1.0 - x) * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0)) <= 2e-30)
		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 (((x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)) <= 2e-30)
		tmp = (1.0 - x) * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], 2e-30], 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-30

   1. Initial program 93.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. lower-+.f64 38.2

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

   5. Applied rewrites 38.2%

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

   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 37.4%

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

   if 2e-30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 87.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 76.0%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.5% 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 89.5%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 53.0%

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

Developer Target 1: 99.5% 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 2024295 
(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)))