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

Percentage Accurate: 89.2% → 98.8%

Time: 13.1s

Alternatives: 15

Speedup: 0.6×

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: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{\mathsf{fma}\left(z, \frac{y}{t\_1}, x + 1\right)}{x + 1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+296}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

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 (- INFINITY))
     t_2
     (if (<= t_3 1e+296)
       t_3
       (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 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= 1e+296) {
		tmp = t_3;
	} 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 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= 1e+296)
		tmp = t_3;
	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, (-Infinity)], t$95$2, If[LessEqual[t$95$3, 1e+296], t$95$3, 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))) < -inf.0 or 9.99999999999999981e295 < (/.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 45.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 99.7%

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

   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 99.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999981e295

   1. Initial program 98.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+296}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ t_3 := t\_1 \cdot \left(x + 1\right)\\ \mathbf{if}\;t\_2 \leq -500000:\\ \;\;\;\;y \cdot \frac{z}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{+296}:\\ \;\;\;\;\frac{y \cdot z}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

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 (* t_1 (+ x 1.0))))
   (if (<= t_2 -500000.0)
     (* y (/ z t_3))
     (if (<= t_2 2e-15)
       (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0))))
       (if (<= t_2 2.0)
         (/ (+ x (/ x (- x (* z t)))) (+ x 1.0))
         (if (<= t_2 1e+296) (/ (* y z) 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 = t_1 * (x + 1.0);
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = y * (z / t_3);
	} else if (t_2 <= 2e-15) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else if (t_2 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_2 <= 1e+296) {
		tmp = (y * z) / t_3;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (z * t) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    t_3 = t_1 * (x + 1.0d0)
    if (t_2 <= (-500000.0d0)) then
        tmp = y * (z / t_3)
    else if (t_2 <= 2d-15) then
        tmp = (x / (x + 1.0d0)) + (y / (t * (x + 1.0d0)))
    else if (t_2 <= 2.0d0) then
        tmp = (x + (x / (x - (z * t)))) / (x + 1.0d0)
    else if (t_2 <= 1d+296) then
        tmp = (y * z) / t_3
    else
        tmp = (x + (y / t)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double t_3 = t_1 * (x + 1.0);
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = y * (z / t_3);
	} else if (t_2 <= 2e-15) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else if (t_2 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_2 <= 1e+296) {
		tmp = (y * z) / 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 = t_1 * (x + 1.0)
	tmp = 0
	if t_2 <= -500000.0:
		tmp = y * (z / t_3)
	elif t_2 <= 2e-15:
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	elif t_2 <= 2.0:
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0)
	elif t_2 <= 1e+296:
		tmp = (y * z) / 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(t_1 * Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -500000.0)
		tmp = Float64(y * Float64(z / t_3));
	elseif (t_2 <= 2e-15)
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(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 <= 1e+296)
		tmp = Float64(Float64(y * z) / 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 = t_1 * (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -500000.0)
		tmp = y * (z / t_3);
	elseif (t_2 <= 2e-15)
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	elseif (t_2 <= 2.0)
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	elseif (t_2 <= 1e+296)
		tmp = (y * z) / 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[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500000.0], N[(y * N[(z / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-15], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $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, 1e+296], N[(N[(y * z), $MachinePrecision] / t$95$3), $MachinePrecision], 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))) < -5e5

   1. Initial program 82.6%

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

      1. lift-+.f64 N/A

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

      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 88.7%

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

   5. Taylor expanded in y around inf

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

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 81.5

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 84.5%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e-15

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.8%

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

   if 2.0000000000000002e-15 < (/.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 99.3

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

   5. Applied rewrites 99.3%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999981e295

   1. Initial program 99.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 9.99999999999999981e295 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 24.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 84.3

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

   5. Applied rewrites 84.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -500000:\\ \;\;\;\;y \cdot \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+296}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ t_3 := t\_1 \cdot \left(x + 1\right)\\ \mathbf{if}\;t\_2 \leq -500000:\\ \;\;\;\;y \cdot \frac{z}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 10^{+296}:\\ \;\;\;\;\frac{y \cdot z}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

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 (* t_1 (+ x 1.0))))
   (if (<= t_2 -500000.0)
     (* y (/ z t_3))
     (if (<= t_2 2e-15)
       (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0))))
       (if (<= t_2 2.0)
         1.0
         (if (<= t_2 1e+296) (/ (* y z) 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 = t_1 * (x + 1.0);
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = y * (z / t_3);
	} else if (t_2 <= 2e-15) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 1e+296) {
		tmp = (y * z) / t_3;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (z * t) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    t_3 = t_1 * (x + 1.0d0)
    if (t_2 <= (-500000.0d0)) then
        tmp = y * (z / t_3)
    else if (t_2 <= 2d-15) then
        tmp = (x / (x + 1.0d0)) + (y / (t * (x + 1.0d0)))
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else if (t_2 <= 1d+296) then
        tmp = (y * z) / t_3
    else
        tmp = (x + (y / t)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double t_3 = t_1 * (x + 1.0);
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = y * (z / t_3);
	} else if (t_2 <= 2e-15) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 1e+296) {
		tmp = (y * z) / 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 = t_1 * (x + 1.0)
	tmp = 0
	if t_2 <= -500000.0:
		tmp = y * (z / t_3)
	elif t_2 <= 2e-15:
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	elif t_2 <= 2.0:
		tmp = 1.0
	elif t_2 <= 1e+296:
		tmp = (y * z) / 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(t_1 * Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -500000.0)
		tmp = Float64(y * Float64(z / t_3));
	elseif (t_2 <= 2e-15)
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * Float64(x + 1.0))));
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 1e+296)
		tmp = Float64(Float64(y * z) / 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 = t_1 * (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -500000.0)
		tmp = y * (z / t_3);
	elseif (t_2 <= 2e-15)
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 1e+296)
		tmp = (y * z) / 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[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500000.0], N[(y * N[(z / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-15], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, 1e+296], N[(N[(y * z), $MachinePrecision] / t$95$3), $MachinePrecision], 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))) < -5e5

   1. Initial program 82.6%

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

      1. lift-+.f64 N/A

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

      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 88.7%

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

   5. Taylor expanded in y around inf

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

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 81.5

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 84.5%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e-15

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.8%

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

   if 2.0000000000000002e-15 < (/.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 99.1%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999981e295

   1. Initial program 99.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 9.99999999999999981e295 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 24.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 84.3

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

   5. Applied rewrites 84.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -500000:\\ \;\;\;\;y \cdot \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+296}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ t_4 := t\_2 \cdot \left(x + 1\right)\\ \mathbf{if}\;t\_3 \leq -500000:\\ \;\;\;\;y \cdot \frac{z}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 10^{+296}:\\ \;\;\;\;\frac{y \cdot z}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* t_2 (+ x 1.0))))
   (if (<= t_3 -500000.0)
     (* y (/ z t_4))
     (if (<= t_3 2e-15)
       t_1
       (if (<= t_3 2.0) 1.0 (if (<= t_3 1e+296) (/ (* y z) 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 = t_2 * (x + 1.0);
	double tmp;
	if (t_3 <= -500000.0) {
		tmp = y * (z / t_4);
	} else if (t_3 <= 2e-15) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= 1e+296) {
		tmp = (y * z) / t_4;
	} 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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (z * t) - x
    t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0d0)
    t_4 = t_2 * (x + 1.0d0)
    if (t_3 <= (-500000.0d0)) then
        tmp = y * (z / t_4)
    else if (t_3 <= 2d-15) then
        tmp = t_1
    else if (t_3 <= 2.0d0) then
        tmp = 1.0d0
    else if (t_3 <= 1d+296) then
        tmp = (y * z) / t_4
    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 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = t_2 * (x + 1.0);
	double tmp;
	if (t_3 <= -500000.0) {
		tmp = y * (z / t_4);
	} else if (t_3 <= 2e-15) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= 1e+296) {
		tmp = (y * z) / 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 = t_2 * (x + 1.0)
	tmp = 0
	if t_3 <= -500000.0:
		tmp = y * (z / t_4)
	elif t_3 <= 2e-15:
		tmp = t_1
	elif t_3 <= 2.0:
		tmp = 1.0
	elif t_3 <= 1e+296:
		tmp = (y * z) / 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(t_2 * Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -500000.0)
		tmp = Float64(y * Float64(z / t_4));
	elseif (t_3 <= 2e-15)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= 1e+296)
		tmp = Float64(Float64(y * z) / 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 = t_2 * (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -500000.0)
		tmp = y * (z / t_4);
	elseif (t_3 <= 2e-15)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= 1e+296)
		tmp = (y * z) / 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[(t$95$2 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -500000.0], N[(y * N[(z / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-15], t$95$1, If[LessEqual[t$95$3, 2.0], 1.0, If[LessEqual[t$95$3, 1e+296], N[(N[(y * z), $MachinePrecision] / t$95$4), $MachinePrecision], 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))) < -5e5

   1. Initial program 82.6%

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

      1. lift-+.f64 N/A

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

      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 88.7%

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

   5. Taylor expanded in y around inf

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

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 81.5

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 84.5%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e-15 or 9.99999999999999981e295 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 2.0000000000000002e-15 < (/.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 99.1%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999981e295

   1. Initial program 99.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -500000:\\ \;\;\;\;y \cdot \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+296}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := z \cdot t - x\\ t_3 := \frac{y \cdot z}{t\_2 \cdot \left(x + 1\right)}\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -500000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_4 \leq 10^{+296}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* z t) x))
        (t_3 (/ (* y z) (* t_2 (+ x 1.0))))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_4 -500000.0)
     t_3
     (if (<= t_4 2e-15)
       t_1
       (if (<= t_4 2.0) 1.0 (if (<= t_4 1e+296) t_3 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 = (y * z) / (t_2 * (x + 1.0));
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -500000.0) {
		tmp = t_3;
	} else if (t_4 <= 2e-15) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = 1.0;
	} else if (t_4 <= 1e+296) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (z * t) - x
    t_3 = (y * z) / (t_2 * (x + 1.0d0))
    t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0d0)
    if (t_4 <= (-500000.0d0)) then
        tmp = t_3
    else if (t_4 <= 2d-15) then
        tmp = t_1
    else if (t_4 <= 2.0d0) then
        tmp = 1.0d0
    else if (t_4 <= 1d+296) then
        tmp = t_3
    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 = (z * t) - x;
	double t_3 = (y * z) / (t_2 * (x + 1.0));
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -500000.0) {
		tmp = t_3;
	} else if (t_4 <= 2e-15) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = 1.0;
	} else if (t_4 <= 1e+296) {
		tmp = t_3;
	} 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 = (y * z) / (t_2 * (x + 1.0))
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -500000.0:
		tmp = t_3
	elif t_4 <= 2e-15:
		tmp = t_1
	elif t_4 <= 2.0:
		tmp = 1.0
	elif t_4 <= 1e+296:
		tmp = t_3
	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(y * z) / Float64(t_2 * Float64(x + 1.0)))
	t_4 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -500000.0)
		tmp = t_3;
	elseif (t_4 <= 2e-15)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = 1.0;
	elseif (t_4 <= 1e+296)
		tmp = t_3;
	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 = (y * z) / (t_2 * (x + 1.0));
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -500000.0)
		tmp = t_3;
	elseif (t_4 <= 2e-15)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = 1.0;
	elseif (t_4 <= 1e+296)
		tmp = t_3;
	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[(y * z), $MachinePrecision] / N[(t$95$2 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -500000.0], t$95$3, If[LessEqual[t$95$4, 2e-15], t$95$1, If[LessEqual[t$95$4, 2.0], 1.0, If[LessEqual[t$95$4, 1e+296], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e5 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999981e295

   1. Initial program 88.7%

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

   3. Taylor expanded in y around inf

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

      1. 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 87.9

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

   5. Applied rewrites 87.9%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e-15 or 9.99999999999999981e295 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 2.0000000000000002e-15 < (/.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 99.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -500000:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+296}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

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-34)
     (/ y (* t (+ x 1.0)))
     (if (<= t_1 2e-60)
       (* x (- 1.0 x))
       (if (<= t_1 2e-15)
         (/ y t)
         (if (<= t_1 1e+28) 1.0 (/ (/ y t) (+ x 1.0))))))))

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-34) {
		tmp = y / (t * (x + 1.0));
	} else if (t_1 <= 2e-60) {
		tmp = x * (1.0 - x);
	} else if (t_1 <= 2e-15) {
		tmp = y / t;
	} else if (t_1 <= 1e+28) {
		tmp = 1.0;
	} else {
		tmp = (y / t) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= (-1d-34)) then
        tmp = y / (t * (x + 1.0d0))
    else if (t_1 <= 2d-60) then
        tmp = x * (1.0d0 - x)
    else if (t_1 <= 2d-15) then
        tmp = y / t
    else if (t_1 <= 1d+28) then
        tmp = 1.0d0
    else
        tmp = (y / t) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e-34) {
		tmp = y / (t * (x + 1.0));
	} else if (t_1 <= 2e-60) {
		tmp = x * (1.0 - x);
	} else if (t_1 <= 2e-15) {
		tmp = y / t;
	} else if (t_1 <= 1e+28) {
		tmp = 1.0;
	} else {
		tmp = (y / t) / (x + 1.0);
	}
	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-34:
		tmp = y / (t * (x + 1.0))
	elif t_1 <= 2e-60:
		tmp = x * (1.0 - x)
	elif t_1 <= 2e-15:
		tmp = y / t
	elif t_1 <= 1e+28:
		tmp = 1.0
	else:
		tmp = (y / t) / (x + 1.0)
	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-34)
		tmp = Float64(y / Float64(t * Float64(x + 1.0)));
	elseif (t_1 <= 2e-60)
		tmp = Float64(x * Float64(1.0 - x));
	elseif (t_1 <= 2e-15)
		tmp = Float64(y / t);
	elseif (t_1 <= 1e+28)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	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-34)
		tmp = y / (t * (x + 1.0));
	elseif (t_1 <= 2e-60)
		tmp = x * (1.0 - x);
	elseif (t_1 <= 2e-15)
		tmp = y / t;
	elseif (t_1 <= 1e+28)
		tmp = 1.0;
	else
		tmp = (y / t) / (x + 1.0);
	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-34], N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-60], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-15], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 1e+28], 1.0, N[(N[(y / t), $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))) < -9.99999999999999928e-35

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      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 y around inf

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

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 52.7%

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

   if -9.99999999999999928e-35 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-60

   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 89.8%

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

   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. +-commutative N/A

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

      3. lower-+.f64 66.1

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

   7. Applied rewrites 66.1%

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

   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 66.1%

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

   if 1.9999999999999999e-60 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e-15

   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 x around 0

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

      1. lower-/.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 2.0000000000000002e-15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999958e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.8%

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

   if 9.99999999999999958e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 49.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 51.5

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

   5. Applied rewrites 51.5%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t \cdot \left(x + 1\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_2 \leq 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* t (+ x 1.0))))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -1e-34)
     t_1
     (if (<= t_2 2e-60)
       (* x (- 1.0 x))
       (if (<= t_2 2e-15) (/ y t) (if (<= t_2 1e+28) 1.0 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (x + 1.0));
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -1e-34) {
		tmp = t_1;
	} else if (t_2 <= 2e-60) {
		tmp = x * (1.0 - x);
	} else if (t_2 <= 2e-15) {
		tmp = y / t;
	} else if (t_2 <= 1e+28) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y / (t * (x + 1.0d0))
    t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_2 <= (-1d-34)) then
        tmp = t_1
    else if (t_2 <= 2d-60) then
        tmp = x * (1.0d0 - x)
    else if (t_2 <= 2d-15) then
        tmp = y / t
    else if (t_2 <= 1d+28) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (x + 1.0));
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -1e-34) {
		tmp = t_1;
	} else if (t_2 <= 2e-60) {
		tmp = x * (1.0 - x);
	} else if (t_2 <= 2e-15) {
		tmp = y / t;
	} else if (t_2 <= 1e+28) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / (t * (x + 1.0))
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= -1e-34:
		tmp = t_1
	elif t_2 <= 2e-60:
		tmp = x * (1.0 - x)
	elif t_2 <= 2e-15:
		tmp = y / t
	elif t_2 <= 1e+28:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(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 <= -1e-34)
		tmp = t_1;
	elseif (t_2 <= 2e-60)
		tmp = Float64(x * Float64(1.0 - x));
	elseif (t_2 <= 2e-15)
		tmp = Float64(y / t);
	elseif (t_2 <= 1e+28)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / (t * (x + 1.0));
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -1e-34)
		tmp = t_1;
	elseif (t_2 <= 2e-60)
		tmp = x * (1.0 - x);
	elseif (t_2 <= 2e-15)
		tmp = y / t;
	elseif (t_2 <= 1e+28)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $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, -1e-34], t$95$1, If[LessEqual[t$95$2, 2e-60], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-15], N[(y / t), $MachinePrecision], If[LessEqual[t$95$2, 1e+28], 1.0, 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))) < -9.99999999999999928e-35 or 9.99999999999999958e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      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 84.0%

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

   5. Taylor expanded in y around inf

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

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 52.0%

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

   if -9.99999999999999928e-35 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-60

   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 89.8%

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

   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. +-commutative N/A

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

      3. lower-+.f64 66.1

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

   7. Applied rewrites 66.1%

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

   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 66.1%

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

   if 1.9999999999999999e-60 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e-15

   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 x around 0

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

      1. lower-/.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 2.0000000000000002e-15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999958e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

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-34)
     (/ y t)
     (if (<= t_1 2e-60)
       (* x (- 1.0 x))
       (if (<= t_1 2e-15) (/ y t) (if (<= t_1 2e+34) 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-34) {
		tmp = y / t;
	} else if (t_1 <= 2e-60) {
		tmp = x * (1.0 - x);
	} else if (t_1 <= 2e-15) {
		tmp = y / t;
	} else if (t_1 <= 2e+34) {
		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-34)) then
        tmp = y / t
    else if (t_1 <= 2d-60) then
        tmp = x * (1.0d0 - x)
    else if (t_1 <= 2d-15) then
        tmp = y / t
    else if (t_1 <= 2d+34) 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-34) {
		tmp = y / t;
	} else if (t_1 <= 2e-60) {
		tmp = x * (1.0 - x);
	} else if (t_1 <= 2e-15) {
		tmp = y / t;
	} else if (t_1 <= 2e+34) {
		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-34:
		tmp = y / t
	elif t_1 <= 2e-60:
		tmp = x * (1.0 - x)
	elif t_1 <= 2e-15:
		tmp = y / t
	elif t_1 <= 2e+34:
		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-34)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-60)
		tmp = Float64(x * Float64(1.0 - x));
	elseif (t_1 <= 2e-15)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e+34)
		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-34)
		tmp = y / t;
	elseif (t_1 <= 2e-60)
		tmp = x * (1.0 - x);
	elseif (t_1 <= 2e-15)
		tmp = y / t;
	elseif (t_1 <= 2e+34)
		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-34], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-60], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-15], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e+34], 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))) < -9.99999999999999928e-35 or 1.9999999999999999e-60 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e-15 or 1.99999999999999989e34 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 49.4

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

   5. Applied rewrites 49.4%

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

   if -9.99999999999999928e-35 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-60

   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 89.8%

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

   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. +-commutative N/A

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

      3. lower-+.f64 66.1

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

   7. Applied rewrites 66.1%

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

   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 66.1%

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

   if 2.0000000000000002e-15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999989e34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{\mathsf{fma}\left(z, \frac{y}{t\_1}, x + 1\right)}{x + 1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -5000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

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 -5000000000000.0)
     t_2
     (if (<= t_3 2e-15)
       (+ (/ x (+ x 1.0)) (/ 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 <= -5000000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 2e-15) {
		tmp = (x / (x + 1.0)) + (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 <= -5000000000000.0)
		tmp = t_2;
	elseif (t_3 <= 2e-15)
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(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, -5000000000000.0], t$95$2, If[LessEqual[t$95$3, 2e-15], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $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))) < -5e12 or 2.0000000000000002e-15 < (/.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 90.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 95.8%

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

   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 95.6%

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

   if -5e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000002e-15

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + 1\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 2e-15) t_1 (if (<= t_2 1e+28) 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 <= 2e-15) {
		tmp = t_1;
	} else if (t_2 <= 1e+28) {
		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 <= 2d-15) then
        tmp = t_1
    else if (t_2 <= 1d+28) 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 <= 2e-15) {
		tmp = t_1;
	} else if (t_2 <= 1e+28) {
		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 <= 2e-15:
		tmp = t_1
	elif t_2 <= 1e+28:
		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 <= 2e-15)
		tmp = t_1;
	elseif (t_2 <= 1e+28)
		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 <= 2e-15)
		tmp = t_1;
	elseif (t_2 <= 1e+28)
		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, 2e-15], t$95$1, If[LessEqual[t$95$2, 1e+28], 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))) < 2.0000000000000002e-15 or 9.99999999999999958e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 76.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if 2.0000000000000002e-15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999958e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0)) <= Inf)
		tmp = Float64(fma(y, Float64(z / Float64(t_1 * Float64(x + 1.0))), Float64(x / Float64(x + 1.0))) + Float64(x / Float64(t_1 * Float64(-1.0 - x))));
	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]}, If[LessEqual[N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(y * N[(z / N[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(t$95$1 * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 97.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.4%

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

Alternative 12: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 2e-15)
     (/ (+ x (/ y t)) 1.0)
     (if (<= t_1 1e+28) 1.0 (/ (/ y t) (+ x 1.0))))))

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 <= 2e-15) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_1 <= 1e+28) {
		tmp = 1.0;
	} else {
		tmp = (y / t) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= 2d-15) then
        tmp = (x + (y / t)) / 1.0d0
    else if (t_1 <= 1d+28) then
        tmp = 1.0d0
    else
        tmp = (y / t) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 2e-15) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_1 <= 1e+28) {
		tmp = 1.0;
	} else {
		tmp = (y / t) / (x + 1.0);
	}
	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 <= 2e-15:
		tmp = (x + (y / t)) / 1.0
	elif t_1 <= 1e+28:
		tmp = 1.0
	else:
		tmp = (y / t) / (x + 1.0)
	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 <= 2e-15)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	elseif (t_1 <= 1e+28)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	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 <= 2e-15)
		tmp = (x + (y / t)) / 1.0;
	elseif (t_1 <= 1e+28)
		tmp = 1.0;
	else
		tmp = (y / t) / (x + 1.0);
	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, 2e-15], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+28], 1.0, N[(N[(y / t), $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))) < 2.0000000000000002e-15

   1. Initial program 90.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.5%

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

   if 2.0000000000000002e-15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999958e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.8%

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

   if 9.99999999999999958e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 49.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 51.5

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

   5. Applied rewrites 51.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;z \leq 500000000000:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t\_1}, x + \frac{x}{x - z \cdot t}\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)))
   (if (<= z -1.55e+114)
     (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0))))
     (if (<= z 500000000000.0)
       (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))
       (/ (fma z (/ y t_1) (+ x (/ x (- x (* z t))))) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double tmp;
	if (z <= -1.55e+114) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else if (z <= 500000000000.0) {
		tmp = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	} else {
		tmp = fma(z, (y / t_1), (x + (x / (x - (z * t))))) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	tmp = 0.0
	if (z <= -1.55e+114)
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * Float64(x + 1.0))));
	elseif (z <= 500000000000.0)
		tmp = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0));
	else
		tmp = Float64(fma(z, Float64(y / t_1), Float64(x + Float64(x / Float64(x - Float64(z * 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]}, If[LessEqual[z, -1.55e+114], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 500000000000.0], N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y / t$95$1), $MachinePrecision] + N[(x + N[(x / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55e114

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 95.9%

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

   if -1.55e114 < z < 5e11

   1. Initial program 99.2%

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

   if 5e11 < z

   1. Initial program 69.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 92.5%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;z \leq 500000000000:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x + \frac{x}{x - z \cdot t}\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)) 2e-60)
   (* 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)) <= 2e-60) {
		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)) <= 2d-60) 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)) <= 2e-60) {
		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)) <= 2e-60:
		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)) <= 2e-60)
		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)) <= 2e-60)
		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], 2e-60], 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))) < 1.9999999999999999e-60

   1. Initial program 89.6%

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

      1. lift-+.f64 N/A

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

      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 89.4%

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

   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. +-commutative N/A

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

      3. lower-+.f64 41.1

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

   7. Applied rewrites 41.1%

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

   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 36.8%

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

   if 1.9999999999999999e-60 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 85.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 73.4%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.9% 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 87.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 50.1%

   \[\leadsto \color{blue}{1} \]
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 2024221 
(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)))