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

Percentage Accurate: 89.3% → 97.3%

Time: 12.4s

Alternatives: 17

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% 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: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{t\_1}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t\_2}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x + \frac{t\_1}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, x, t\right)} + \left(\frac{x}{x + 1} - \frac{x}{t \cdot \mathsf{fma}\left(x, z, z\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) x))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ t_1 t_2)) (+ x 1.0))))
   (if (<= t_3 (- INFINITY))
     (/ (fma z (/ y t_2) (+ x 1.0)) (+ x 1.0))
     (if (<= t_3 2e+294)
       (/ (+ x (/ t_1 (fma z t (- x)))) (+ x 1.0))
       (+ (/ y (fma t x t)) (- (/ x (+ x 1.0)) (/ x (* t (fma x z z)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (z * t) - x;
	double t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(z, (y / t_2), (x + 1.0)) / (x + 1.0);
	} else if (t_3 <= 2e+294) {
		tmp = (x + (t_1 / fma(z, t, -x))) / (x + 1.0);
	} else {
		tmp = (y / fma(t, x, t)) + ((x / (x + 1.0)) - (x / (t * fma(x, z, z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - x)
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(t_1 / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(fma(z, Float64(y / t_2), Float64(x + 1.0)) / Float64(x + 1.0));
	elseif (t_3 <= 2e+294)
		tmp = Float64(Float64(x + Float64(t_1 / fma(z, t, Float64(-x)))) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(y / fma(t, x, t)) + Float64(Float64(x / Float64(x + 1.0)) - Float64(x / Float64(t * fma(x, z, z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(z * N[(y / t$95$2), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+294], N[(N[(x + N[(t$95$1 / N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision] + N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(t * N[(x * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 17.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 93.0%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e294

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 13.8%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-rgt-in N/A

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

      17. *-lft-identity N/A

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

      18. lower-fma.f64 84.4

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

   5. Applied rewrites 84.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, x, t\right)} + \left(\frac{x}{x + 1} - \frac{x}{t \cdot \mathsf{fma}\left(x, z, z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ t_3 := \frac{y \cdot z}{t\_1 \cdot \left(x + 1\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.8:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0)))
        (t_3 (/ (* y z) (* t_1 (+ x 1.0)))))
   (if (<= t_2 (- INFINITY))
     (/ (+ x (* y (/ 1.0 t))) (+ x 1.0))
     (if (<= t_2 -2e+14)
       t_3
       (if (<= t_2 0.8)
         (/ (- x (/ (- (/ x z) y) t)) (+ x 1.0))
         (if (<= t_2 2.0)
           (/ (+ x (/ x (- x (* z t)))) (+ x 1.0))
           (if (<= t_2 2e+294) t_3 (/ (+ x (/ y t)) (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double t_3 = (y * z) / (t_1 * (x + 1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_2 <= -2e+14) {
		tmp = t_3;
	} else if (t_2 <= 0.8) {
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0);
	} else if (t_2 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_2 <= 2e+294) {
		tmp = t_3;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double t_3 = (y * z) / (t_1 * (x + 1.0));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_2 <= -2e+14) {
		tmp = t_3;
	} else if (t_2 <= 0.8) {
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0);
	} else if (t_2 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_2 <= 2e+294) {
		tmp = t_3;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	t_3 = (y * z) / (t_1 * (x + 1.0))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (x + (y * (1.0 / t))) / (x + 1.0)
	elif t_2 <= -2e+14:
		tmp = t_3
	elif t_2 <= 0.8:
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0)
	elif t_2 <= 2.0:
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0)
	elif t_2 <= 2e+294:
		tmp = t_3
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	t_3 = Float64(Float64(y * z) / Float64(t_1 * Float64(x + 1.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(x + Float64(y * Float64(1.0 / t))) / Float64(x + 1.0));
	elseif (t_2 <= -2e+14)
		tmp = t_3;
	elseif (t_2 <= 0.8)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x + 1.0));
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x + Float64(x / Float64(x - Float64(z * t)))) / Float64(x + 1.0));
	elseif (t_2 <= 2e+294)
		tmp = t_3;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	t_3 = (y * z) / (t_1 * (x + 1.0));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	elseif (t_2 <= -2e+14)
		tmp = t_3;
	elseif (t_2 <= 0.8)
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0);
	elseif (t_2 <= 2.0)
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	elseif (t_2 <= 2e+294)
		tmp = t_3;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] / N[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(x + N[(y * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+14], t$95$3, If[LessEqual[t$95$2, 0.8], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x + N[(x / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+294], t$95$3, N[(N[(x + N[(y / t), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 17.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e294

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.80000000000000004

   1. Initial program 95.9%

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

   3. Taylor expanded in t around -inf

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 0.80000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 13.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.8:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ t_3 := \frac{y \cdot z}{t\_1 \cdot \left(x + 1\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0)))
        (t_3 (/ (* y z) (* t_1 (+ x 1.0)))))
   (if (<= t_2 (- INFINITY))
     (/ (+ x (* y (/ 1.0 t))) (+ x 1.0))
     (if (<= t_2 -2e+14)
       t_3
       (if (<= t_2 2e-16)
         (/ (- x (/ (- (/ x z) y) t)) 1.0)
         (if (<= t_2 2.0)
           (/ (+ x (/ x (- x (* z t)))) (+ x 1.0))
           (if (<= t_2 2e+294) t_3 (/ (+ x (/ y t)) (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double t_3 = (y * z) / (t_1 * (x + 1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_2 <= -2e+14) {
		tmp = t_3;
	} else if (t_2 <= 2e-16) {
		tmp = (x - (((x / z) - y) / t)) / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_2 <= 2e+294) {
		tmp = t_3;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double t_3 = (y * z) / (t_1 * (x + 1.0));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_2 <= -2e+14) {
		tmp = t_3;
	} else if (t_2 <= 2e-16) {
		tmp = (x - (((x / z) - y) / t)) / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_2 <= 2e+294) {
		tmp = t_3;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	t_3 = (y * z) / (t_1 * (x + 1.0))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (x + (y * (1.0 / t))) / (x + 1.0)
	elif t_2 <= -2e+14:
		tmp = t_3
	elif t_2 <= 2e-16:
		tmp = (x - (((x / z) - y) / t)) / 1.0
	elif t_2 <= 2.0:
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0)
	elif t_2 <= 2e+294:
		tmp = t_3
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	t_3 = Float64(Float64(y * z) / Float64(t_1 * Float64(x + 1.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(x + Float64(y * Float64(1.0 / t))) / Float64(x + 1.0));
	elseif (t_2 <= -2e+14)
		tmp = t_3;
	elseif (t_2 <= 2e-16)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / 1.0);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x + Float64(x / Float64(x - Float64(z * t)))) / Float64(x + 1.0));
	elseif (t_2 <= 2e+294)
		tmp = t_3;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	t_3 = (y * z) / (t_1 * (x + 1.0));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	elseif (t_2 <= -2e+14)
		tmp = t_3;
	elseif (t_2 <= 2e-16)
		tmp = (x - (((x / z) - y) / t)) / 1.0;
	elseif (t_2 <= 2.0)
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	elseif (t_2 <= 2e+294)
		tmp = t_3;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] / N[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(x + N[(y * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+14], t$95$3, If[LessEqual[t$95$2, 2e-16], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x + N[(x / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+294], t$95$3, N[(N[(x + N[(y / t), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 17.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e294

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-16

   1. Initial program 95.8%

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

   3. Taylor expanded in t around -inf

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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 2e-16 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 13.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{t\_1}{t\_2}}{x + 1}\\ t_4 := \frac{y \cdot z}{t\_2 \cdot \left(x + 1\right)}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x + \frac{t\_1}{z \cdot t}}{1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) x))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ t_1 t_2)) (+ x 1.0)))
        (t_4 (/ (* y z) (* t_2 (+ x 1.0)))))
   (if (<= t_3 (- INFINITY))
     (/ (+ x (* y (/ 1.0 t))) (+ x 1.0))
     (if (<= t_3 -2e+14)
       t_4
       (if (<= t_3 2e-16)
         (/ (+ x (/ t_1 (* z t))) 1.0)
         (if (<= t_3 2.0)
           (/ (+ x (/ x (- x (* z t)))) (+ x 1.0))
           (if (<= t_3 2e+294) t_4 (/ (+ x (/ y t)) (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (z * t) - x;
	double t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	double t_4 = (y * z) / (t_2 * (x + 1.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_3 <= -2e+14) {
		tmp = t_4;
	} else if (t_3 <= 2e-16) {
		tmp = (x + (t_1 / (z * t))) / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_3 <= 2e+294) {
		tmp = t_4;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (z * t) - x;
	double t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	double t_4 = (y * z) / (t_2 * (x + 1.0));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_3 <= -2e+14) {
		tmp = t_4;
	} else if (t_3 <= 2e-16) {
		tmp = (x + (t_1 / (z * t))) / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_3 <= 2e+294) {
		tmp = t_4;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) - x
	t_2 = (z * t) - x
	t_3 = (x + (t_1 / t_2)) / (x + 1.0)
	t_4 = (y * z) / (t_2 * (x + 1.0))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (x + (y * (1.0 / t))) / (x + 1.0)
	elif t_3 <= -2e+14:
		tmp = t_4
	elif t_3 <= 2e-16:
		tmp = (x + (t_1 / (z * t))) / 1.0
	elif t_3 <= 2.0:
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0)
	elif t_3 <= 2e+294:
		tmp = t_4
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - x)
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(t_1 / t_2)) / Float64(x + 1.0))
	t_4 = Float64(Float64(y * z) / Float64(t_2 * Float64(x + 1.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(x + Float64(y * Float64(1.0 / t))) / Float64(x + 1.0));
	elseif (t_3 <= -2e+14)
		tmp = t_4;
	elseif (t_3 <= 2e-16)
		tmp = Float64(Float64(x + Float64(t_1 / Float64(z * t))) / 1.0);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x + Float64(x / Float64(x - Float64(z * t)))) / Float64(x + 1.0));
	elseif (t_3 <= 2e+294)
		tmp = t_4;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) - x;
	t_2 = (z * t) - x;
	t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	t_4 = (y * z) / (t_2 * (x + 1.0));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	elseif (t_3 <= -2e+14)
		tmp = t_4;
	elseif (t_3 <= 2e-16)
		tmp = (x + (t_1 / (z * t))) / 1.0;
	elseif (t_3 <= 2.0)
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	elseif (t_3 <= 2e+294)
		tmp = t_4;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * z), $MachinePrecision] / N[(t$95$2 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(x + N[(y * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e+14], t$95$4, If[LessEqual[t$95$3, 2e-16], N[(N[(x + N[(t$95$1 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x + N[(x / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+294], t$95$4, N[(N[(x + N[(y / t), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 17.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e294

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-16

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.4%

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

   9. Taylor expanded in t around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 95.7

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

   11. Applied rewrites 95.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 13.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ t_4 := \frac{y \cdot z}{t\_2 \cdot \left(x + 1\right)}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0)))
        (t_4 (/ (* y z) (* t_2 (+ x 1.0)))))
   (if (<= t_3 (- INFINITY))
     (/ (+ x (* y (/ 1.0 t))) (+ x 1.0))
     (if (<= t_3 -2e+14)
       t_4
       (if (<= t_3 2e-30)
         t_1
         (if (<= t_3 2.0)
           (/ (+ x (/ x (- x (* z t)))) (+ x 1.0))
           (if (<= t_3 2e+294) t_4 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (y * z) / (t_2 * (x + 1.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_3 <= -2e+14) {
		tmp = t_4;
	} else if (t_3 <= 2e-30) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_3 <= 2e+294) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (y * z) / (t_2 * (x + 1.0));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_3 <= -2e+14) {
		tmp = t_4;
	} else if (t_3 <= 2e-30) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_3 <= 2e+294) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (z * t) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	t_4 = (y * z) / (t_2 * (x + 1.0))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (x + (y * (1.0 / t))) / (x + 1.0)
	elif t_3 <= -2e+14:
		tmp = t_4
	elif t_3 <= 2e-30:
		tmp = t_1
	elif t_3 <= 2.0:
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0)
	elif t_3 <= 2e+294:
		tmp = t_4
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	t_4 = Float64(Float64(y * z) / Float64(t_2 * Float64(x + 1.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(x + Float64(y * Float64(1.0 / t))) / Float64(x + 1.0));
	elseif (t_3 <= -2e+14)
		tmp = t_4;
	elseif (t_3 <= 2e-30)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x + Float64(x / Float64(x - Float64(z * t)))) / Float64(x + 1.0));
	elseif (t_3 <= 2e+294)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (z * t) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	t_4 = (y * z) / (t_2 * (x + 1.0));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	elseif (t_3 <= -2e+14)
		tmp = t_4;
	elseif (t_3 <= 2e-30)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	elseif (t_3 <= 2e+294)
		tmp = t_4;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * z), $MachinePrecision] / N[(t$95$2 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(x + N[(y * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e+14], t$95$4, If[LessEqual[t$95$3, 2e-30], t$95$1, If[LessEqual[t$95$3, 2.0], N[(N[(x + N[(x / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+294], t$95$4, t$95$1]]]]]]]]]

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 17.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e294

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-30 or 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

   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 y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ t_4 := \frac{y \cdot z}{t\_2 \cdot \left(x + 1\right)}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0)))
        (t_4 (/ (* y z) (* t_2 (+ x 1.0)))))
   (if (<= t_3 (- INFINITY))
     (/ (+ x (* y (/ 1.0 t))) (+ x 1.0))
     (if (<= t_3 -2e+14)
       t_4
       (if (<= t_3 0.8)
         t_1
         (if (<= t_3 2.0) 1.0 (if (<= t_3 2e+294) t_4 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (y * z) / (t_2 * (x + 1.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_3 <= -2e+14) {
		tmp = t_4;
	} else if (t_3 <= 0.8) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= 2e+294) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (y * z) / (t_2 * (x + 1.0));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	} else if (t_3 <= -2e+14) {
		tmp = t_4;
	} else if (t_3 <= 0.8) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= 2e+294) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (z * t) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	t_4 = (y * z) / (t_2 * (x + 1.0))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (x + (y * (1.0 / t))) / (x + 1.0)
	elif t_3 <= -2e+14:
		tmp = t_4
	elif t_3 <= 0.8:
		tmp = t_1
	elif t_3 <= 2.0:
		tmp = 1.0
	elif t_3 <= 2e+294:
		tmp = t_4
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	t_4 = Float64(Float64(y * z) / Float64(t_2 * Float64(x + 1.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(x + Float64(y * Float64(1.0 / t))) / Float64(x + 1.0));
	elseif (t_3 <= -2e+14)
		tmp = t_4;
	elseif (t_3 <= 0.8)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= 2e+294)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (z * t) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	t_4 = (y * z) / (t_2 * (x + 1.0));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (x + (y * (1.0 / t))) / (x + 1.0);
	elseif (t_3 <= -2e+14)
		tmp = t_4;
	elseif (t_3 <= 0.8)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= 2e+294)
		tmp = t_4;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * z), $MachinePrecision] / N[(t$95$2 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(x + N[(y * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e+14], t$95$4, If[LessEqual[t$95$3, 0.8], t$95$1, If[LessEqual[t$95$3, 2.0], 1.0, If[LessEqual[t$95$3, 2e+294], t$95$4, t$95$1]]]]]]]]]

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 17.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e294

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.80000000000000004 or 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 68.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if 0.80000000000000004 < (/.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 97.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{1}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.8:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ t_4 := \frac{y \cdot z}{t\_2 \cdot \left(x + 1\right)}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0)))
        (t_4 (/ (* y z) (* t_2 (+ x 1.0)))))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -2e+14)
       t_4
       (if (<= t_3 0.8)
         t_1
         (if (<= t_3 2.0) 1.0 (if (<= t_3 2e+294) t_4 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (y * z) / (t_2 * (x + 1.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -2e+14) {
		tmp = t_4;
	} else if (t_3 <= 0.8) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= 2e+294) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (y * z) / (t_2 * (x + 1.0));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_3 <= -2e+14) {
		tmp = t_4;
	} else if (t_3 <= 0.8) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= 2e+294) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (z * t) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	t_4 = (y * z) / (t_2 * (x + 1.0))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_1
	elif t_3 <= -2e+14:
		tmp = t_4
	elif t_3 <= 0.8:
		tmp = t_1
	elif t_3 <= 2.0:
		tmp = 1.0
	elif t_3 <= 2e+294:
		tmp = t_4
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	t_4 = Float64(Float64(y * z) / Float64(t_2 * Float64(x + 1.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -2e+14)
		tmp = t_4;
	elseif (t_3 <= 0.8)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= 2e+294)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (z * t) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	t_4 = (y * z) / (t_2 * (x + 1.0));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_1;
	elseif (t_3 <= -2e+14)
		tmp = t_4;
	elseif (t_3 <= 0.8)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= 2e+294)
		tmp = t_4;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * z), $MachinePrecision] / N[(t$95$2 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -2e+14], t$95$4, If[LessEqual[t$95$3, 0.8], t$95$1, If[LessEqual[t$95$3, 2.0], 1.0, If[LessEqual[t$95$3, 2e+294], t$95$4, 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))) < -inf.0 or -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.80000000000000004 or 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 60.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e294

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.80000000000000004 < (/.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 97.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.8:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, t, -x\right)}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 0.8:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t\_1}, x + 1\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -2e+14)
     (/ (fma z (/ y (fma z t (- x))) (+ x 1.0)) (+ x 1.0))
     (if (<= t_2 0.8)
       (/ (- x (/ (- (/ x z) y) t)) (+ x 1.0))
       (if (<= t_2 INFINITY)
         (/ (fma z (/ y t_1) (+ x 1.0)) (+ x 1.0))
         (/ (+ x (/ y t)) (+ x 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+14) {
		tmp = fma(z, (y / fma(z, t, -x)), (x + 1.0)) / (x + 1.0);
	} else if (t_2 <= 0.8) {
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma(z, (y / t_1), (x + 1.0)) / (x + 1.0);
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e+14)
		tmp = Float64(fma(z, Float64(y / fma(z, t, Float64(-x))), Float64(x + 1.0)) / Float64(x + 1.0));
	elseif (t_2 <= 0.8)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x + 1.0));
	elseif (t_2 <= Inf)
		tmp = Float64(fma(z, Float64(y / t_1), Float64(x + 1.0)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+14], N[(N[(z * N[(y / N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.8], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(z * N[(y / t$95$1), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 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))) < -2e14

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 87.6

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

   9. Applied rewrites 87.6%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.80000000000000004

   1. Initial program 95.9%

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

   3. Taylor expanded in t around -inf

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 0.80000000000000004 < (/.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 95.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 96.7%

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

   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

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(z, t, -x\right)}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.8:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{\mathsf{fma}\left(z, \frac{y}{t\_1}, x + 1\right)}{x + 1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.8:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (/ (fma z (/ y t_1) (+ x 1.0)) (+ x 1.0)))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_3 -2e+14)
     t_2
     (if (<= t_3 0.8)
       (/ (- x (/ (- (/ x z) y) t)) (+ x 1.0))
       (if (<= t_3 INFINITY) t_2 (/ (+ x (/ y t)) (+ x 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = fma(z, (y / t_1), (x + 1.0)) / (x + 1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -2e+14) {
		tmp = t_2;
	} else if (t_3 <= 0.8) {
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(fma(z, Float64(y / t_1), Float64(x + 1.0)) / Float64(x + 1.0))
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -2e+14)
		tmp = t_2;
	elseif (t_3 <= 0.8)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x + 1.0));
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * N[(y / t$95$1), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+14], t$95$2, If[LessEqual[t$95$3, 0.8], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(N[(x + N[(y / t), $MachinePrecision]), $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))) < -2e14 or 0.80000000000000004 < (/.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 89.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 94.8%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.80000000000000004

   1. Initial program 95.9%

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

   3. Taylor expanded in t around -inf

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.8:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{-48}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -1e-32)
     (/ (/ y t) (+ x 1.0))
     (if (<= t_1 1e-48) (* x (- 1.0 x)) (if (<= t_1 1e+71) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e-32) {
		tmp = (y / t) / (x + 1.0);
	} else if (t_1 <= 1e-48) {
		tmp = x * (1.0 - x);
	} else if (t_1 <= 1e+71) {
		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 + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= (-1d-32)) then
        tmp = (y / t) / (x + 1.0d0)
    else if (t_1 <= 1d-48) then
        tmp = x * (1.0d0 - x)
    else if (t_1 <= 1d+71) 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 + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e-32) {
		tmp = (y / t) / (x + 1.0);
	} else if (t_1 <= 1e-48) {
		tmp = x * (1.0 - x);
	} else if (t_1 <= 1e+71) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -1e-32:
		tmp = (y / t) / (x + 1.0)
	elif t_1 <= 1e-48:
		tmp = x * (1.0 - x)
	elif t_1 <= 1e+71:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e-32)
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	elseif (t_1 <= 1e-48)
		tmp = Float64(x * Float64(1.0 - x));
	elseif (t_1 <= 1e+71)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -1e-32)
		tmp = (y / t) / (x + 1.0);
	elseif (t_1 <= 1e-48)
		tmp = x * (1.0 - x);
	elseif (t_1 <= 1e+71)
		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[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-32], N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-48], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+71], 1.0, N[(y / t), $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))) < -1.00000000000000006e-32

   1. Initial program 73.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

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

   5. Applied rewrites 51.2%

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

   if -1.00000000000000006e-32 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999997e-49

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 65.2

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

   7. Applied rewrites 65.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.2%

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

   if 9.9999999999999997e-49 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e71

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 90.0%

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

   if 1e71 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 44.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 43.0

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

   5. Applied rewrites 43.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{-48}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 10^{-48}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -1e-32)
     (/ y t)
     (if (<= t_1 1e-48) (* x (- 1.0 x)) (if (<= t_1 1e+71) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e-32) {
		tmp = y / t;
	} else if (t_1 <= 1e-48) {
		tmp = x * (1.0 - x);
	} else if (t_1 <= 1e+71) {
		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 + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= (-1d-32)) then
        tmp = y / t
    else if (t_1 <= 1d-48) then
        tmp = x * (1.0d0 - x)
    else if (t_1 <= 1d+71) 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 + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e-32) {
		tmp = y / t;
	} else if (t_1 <= 1e-48) {
		tmp = x * (1.0 - x);
	} else if (t_1 <= 1e+71) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -1e-32:
		tmp = y / t
	elif t_1 <= 1e-48:
		tmp = x * (1.0 - x)
	elif t_1 <= 1e+71:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e-32)
		tmp = Float64(y / t);
	elseif (t_1 <= 1e-48)
		tmp = Float64(x * Float64(1.0 - x));
	elseif (t_1 <= 1e+71)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -1e-32)
		tmp = y / t;
	elseif (t_1 <= 1e-48)
		tmp = x * (1.0 - x);
	elseif (t_1 <= 1e+71)
		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[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-32], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 1e-48], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+71], 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))) < -1.00000000000000006e-32 or 1e71 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 60.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 45.0

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

   5. Applied rewrites 45.0%

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

   if -1.00000000000000006e-32 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999997e-49

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 65.2

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

   7. Applied rewrites 65.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.2%

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

   if 9.9999999999999997e-49 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e71

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 90.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{-48}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{t\_1}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t\_2}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x + \frac{t\_1}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) x))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ t_1 t_2)) (+ x 1.0))))
   (if (<= t_3 (- INFINITY))
     (/ (fma z (/ y t_2) (+ x 1.0)) (+ x 1.0))
     (if (<= t_3 2e+294)
       (/ (+ x (/ t_1 (fma z t (- x)))) (+ x 1.0))
       (/ (+ x (/ y t)) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (z * t) - x;
	double t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(z, (y / t_2), (x + 1.0)) / (x + 1.0);
	} else if (t_3 <= 2e+294) {
		tmp = (x + (t_1 / fma(z, t, -x))) / (x + 1.0);
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - x)
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(t_1 / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(fma(z, Float64(y / t_2), Float64(x + 1.0)) / Float64(x + 1.0));
	elseif (t_3 <= 2e+294)
		tmp = Float64(Float64(x + Float64(t_1 / fma(z, t, Float64(-x)))) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(z * N[(y / t$95$2), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+294], N[(N[(x + N[(t$95$1 / N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y / t), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 17.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 93.0%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e294

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 13.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t\_1}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 (- INFINITY))
     (/ (fma z (/ y t_1) (+ x 1.0)) (+ x 1.0))
     (if (<= t_2 2e+294) t_2 (/ (+ x (/ y t)) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(z, (y / t_1), (x + 1.0)) / (x + 1.0);
	} else if (t_2 <= 2e+294) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(z, Float64(y / t_1), Float64(x + 1.0)) / Float64(x + 1.0));
	elseif (t_2 <= 2e+294)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(z * N[(y / t$95$1), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+294], t$95$2, N[(N[(x + N[(y / t), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 17.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 93.0%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e294

   1. Initial program 98.9%

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

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 13.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 0.8) t_1 (if (<= t_2 1e+71) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.8) {
		tmp = t_1;
	} else if (t_2 <= 1e+71) {
		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 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_2 <= 0.8d0) then
        tmp = t_1
    else if (t_2 <= 1d+71) 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 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.8) {
		tmp = t_1;
	} else if (t_2 <= 1e+71) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= 0.8:
		tmp = t_1
	elif t_2 <= 1e+71:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 0.8)
		tmp = t_1;
	elseif (t_2 <= 1e+71)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 0.8)
		tmp = t_1;
	elseif (t_2 <= 1e+71)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.8], t$95$1, If[LessEqual[t$95$2, 1e+71], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.80000000000000004 or 1e71 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if 0.80000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e71

   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 93.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.8:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) 1.0))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 2e-16) t_1 (if (<= t_2 1e+71) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / 1.0;
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 2e-16) {
		tmp = t_1;
	} else if (t_2 <= 1e+71) {
		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 = (x + (y / t)) / 1.0d0
    t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_2 <= 2d-16) then
        tmp = t_1
    else if (t_2 <= 1d+71) 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 = (x + (y / t)) / 1.0;
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 2e-16) {
		tmp = t_1;
	} else if (t_2 <= 1e+71) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / 1.0
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= 2e-16:
		tmp = t_1
	elif t_2 <= 1e+71:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / 1.0)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 2e-16)
		tmp = t_1;
	elseif (t_2 <= 1e+71)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / 1.0;
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 2e-16)
		tmp = t_1;
	elseif (t_2 <= 1e+71)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-16], t$95$1, If[LessEqual[t$95$2, 1e+71], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-16 or 1e71 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.7%

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

   if 2e-16 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e71

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 92.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{-48}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)) 1e-48)
   (* x (- 1.0 x))
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 1e-48) {
		tmp = x * (1.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)) <= 1d-48) then
        tmp = x * (1.0d0 - x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 1e-48) {
		tmp = x * (1.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 1e-48:
		tmp = x * (1.0 - x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0)) <= 1e-48)
		tmp = Float64(x * Float64(1.0 - x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 1e-48)
		tmp = x * (1.0 - x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1e-48], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999997e-49

   1. Initial program 83.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 33.0%

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

   if 9.9999999999999997e-49 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   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 x around inf

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

   5. Applied rewrites 74.4%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{-48}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.4% 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 86.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 52.6%

   \[\leadsto \color{blue}{1} \]
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 2024232 
(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)))