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

Percentage Accurate: 88.7% → 97.2%

Time: 10.1s

Alternatives: 18

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - t \cdot z\\ t_2 := t \cdot z - x\\ t_3 := \frac{x - \frac{x - z \cdot y}{t\_2}}{x - -1}\\ \mathbf{if}\;t\_3 \leq -500000000000:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{t\_1} \cdot y\\ \mathbf{elif}\;t\_3 \leq 0.002:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x - -1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x - -1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{y}{\frac{-1 - x}{z} \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (* t z)))
        (t_2 (- (* t z) x))
        (t_3 (/ (- x (/ (- x (* z y)) t_2)) (- x -1.0))))
   (if (<= t_3 -500000000000.0)
     (* (/ (/ z (- -1.0 x)) t_1) y)
     (if (<= t_3 0.002)
       (/ (- x (/ (- (/ x z) y) t)) (- x -1.0))
       (if (<= t_3 2.0)
         (/ (- x (/ x t_2)) (- x -1.0))
         (if (<= t_3 INFINITY)
           (/ y (* (/ (- -1.0 x) z) t_1))
           (/ (+ (/ y t) x) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (t * z);
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double tmp;
	if (t_3 <= -500000000000.0) {
		tmp = ((z / (-1.0 - x)) / t_1) * y;
	} else if (t_3 <= 0.002) {
		tmp = (x - (((x / z) - y) / t)) / (x - -1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = y / (((-1.0 - x) / z) * t_1);
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (t * z);
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double tmp;
	if (t_3 <= -500000000000.0) {
		tmp = ((z / (-1.0 - x)) / t_1) * y;
	} else if (t_3 <= 0.002) {
		tmp = (x - (((x / z) - y) / t)) / (x - -1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = y / (((-1.0 - x) / z) * t_1);
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (t * z)
	t_2 = (t * z) - x
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0)
	tmp = 0
	if t_3 <= -500000000000.0:
		tmp = ((z / (-1.0 - x)) / t_1) * y
	elif t_3 <= 0.002:
		tmp = (x - (((x / z) - y) / t)) / (x - -1.0)
	elif t_3 <= 2.0:
		tmp = (x - (x / t_2)) / (x - -1.0)
	elif t_3 <= math.inf:
		tmp = y / (((-1.0 - x) / z) * t_1)
	else:
		tmp = ((y / t) + x) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(t * z))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / t_2)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_3 <= -500000000000.0)
		tmp = Float64(Float64(Float64(z / Float64(-1.0 - x)) / t_1) * y);
	elseif (t_3 <= 0.002)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x - -1.0));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x - -1.0));
	elseif (t_3 <= Inf)
		tmp = Float64(y / Float64(Float64(Float64(-1.0 - x) / z) * t_1));
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (t * z);
	t_2 = (t * z) - x;
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	tmp = 0.0;
	if (t_3 <= -500000000000.0)
		tmp = ((z / (-1.0 - x)) / t_1) * y;
	elseif (t_3 <= 0.002)
		tmp = (x - (((x / z) - y) / t)) / (x - -1.0);
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_2)) / (x - -1.0);
	elseif (t_3 <= Inf)
		tmp = y / (((-1.0 - x) / z) * t_1);
	else
		tmp = ((y / t) + x) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -500000000000.0], N[(N[(N[(z / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$3, 0.002], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(y / N[(N[(N[(-1.0 - x), $MachinePrecision] / z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 73.6%

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

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.2

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 95.2%

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

   if -5e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-3

   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 t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 2e-3 < (/.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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   7. 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 99.5

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

   8. Applied rewrites 99.5%

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

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

   1. Initial program 89.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 99.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -500000000000:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.002:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{y}{\frac{-1 - x}{z} \cdot \left(x - t \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ t_2 := t \cdot z - x\\ t_3 := \frac{x - \frac{x - z \cdot y}{t\_2}}{x - -1}\\ \mathbf{if}\;t\_3 \leq -500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.002:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x - -1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x - -1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (/ z (- -1.0 x)) (- x (* t z))) y))
        (t_2 (- (* t z) x))
        (t_3 (/ (- x (/ (- x (* z y)) t_2)) (- x -1.0))))
   (if (<= t_3 -500000000000.0)
     t_1
     (if (<= t_3 0.002)
       (/ (- x (/ (- (/ x z) y) t)) (- x -1.0))
       (if (<= t_3 2.0)
         (/ (- x (/ x t_2)) (- x -1.0))
         (if (<= t_3 INFINITY) t_1 (/ (+ (/ y t) x) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double tmp;
	if (t_3 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_3 <= 0.002) {
		tmp = (x - (((x / z) - y) / t)) / (x - -1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double tmp;
	if (t_3 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_3 <= 0.002) {
		tmp = (x - (((x / z) - y) / t)) / (x - -1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y
	t_2 = (t * z) - x
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0)
	tmp = 0
	if t_3 <= -500000000000.0:
		tmp = t_1
	elif t_3 <= 0.002:
		tmp = (x - (((x / z) - y) / t)) / (x - -1.0)
	elif t_3 <= 2.0:
		tmp = (x - (x / t_2)) / (x - -1.0)
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z / Float64(-1.0 - x)) / Float64(x - Float64(t * z))) * y)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / t_2)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_3 <= -500000000000.0)
		tmp = t_1;
	elseif (t_3 <= 0.002)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x - -1.0));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x - -1.0));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	t_2 = (t * z) - x;
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	tmp = 0.0;
	if (t_3 <= -500000000000.0)
		tmp = t_1;
	elseif (t_3 <= 0.002)
		tmp = (x - (((x / z) - y) / t)) / (x - -1.0);
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_2)) / (x - -1.0);
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -500000000000.0], t$95$1, If[LessEqual[t$95$3, 0.002], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e11 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 82.8

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 97.0%

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

   if -5e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-3

   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 t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 2e-3 < (/.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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   7. 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 99.5

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

   8. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -500000000000:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.002:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ t_2 := x - z \cdot y\\ t_3 := t \cdot z - x\\ t_4 := \frac{x - \frac{t\_2}{t\_3}}{x - -1}\\ \mathbf{if}\;t\_4 \leq -500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.002:\\ \;\;\;\;\frac{x - \frac{t\_2}{t \cdot z}}{x - -1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_3}}{x - -1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (/ z (- -1.0 x)) (- x (* t z))) y))
        (t_2 (- x (* z y)))
        (t_3 (- (* t z) x))
        (t_4 (/ (- x (/ t_2 t_3)) (- x -1.0))))
   (if (<= t_4 -500000000000.0)
     t_1
     (if (<= t_4 0.002)
       (/ (- x (/ t_2 (* t z))) (- x -1.0))
       (if (<= t_4 2.0)
         (/ (- x (/ x t_3)) (- x -1.0))
         (if (<= t_4 INFINITY) t_1 (/ (+ (/ y t) x) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = x - (z * y);
	double t_3 = (t * z) - x;
	double t_4 = (x - (t_2 / t_3)) / (x - -1.0);
	double tmp;
	if (t_4 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_4 <= 0.002) {
		tmp = (x - (t_2 / (t * z))) / (x - -1.0);
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_3)) / (x - -1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = x - (z * y);
	double t_3 = (t * z) - x;
	double t_4 = (x - (t_2 / t_3)) / (x - -1.0);
	double tmp;
	if (t_4 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_4 <= 0.002) {
		tmp = (x - (t_2 / (t * z))) / (x - -1.0);
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_3)) / (x - -1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y
	t_2 = x - (z * y)
	t_3 = (t * z) - x
	t_4 = (x - (t_2 / t_3)) / (x - -1.0)
	tmp = 0
	if t_4 <= -500000000000.0:
		tmp = t_1
	elif t_4 <= 0.002:
		tmp = (x - (t_2 / (t * z))) / (x - -1.0)
	elif t_4 <= 2.0:
		tmp = (x - (x / t_3)) / (x - -1.0)
	elif t_4 <= math.inf:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z / Float64(-1.0 - x)) / Float64(x - Float64(t * z))) * y)
	t_2 = Float64(x - Float64(z * y))
	t_3 = Float64(Float64(t * z) - x)
	t_4 = Float64(Float64(x - Float64(t_2 / t_3)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_4 <= -500000000000.0)
		tmp = t_1;
	elseif (t_4 <= 0.002)
		tmp = Float64(Float64(x - Float64(t_2 / Float64(t * z))) / Float64(x - -1.0));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_3)) / Float64(x - -1.0));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	t_2 = x - (z * y);
	t_3 = (t * z) - x;
	t_4 = (x - (t_2 / t_3)) / (x - -1.0);
	tmp = 0.0;
	if (t_4 <= -500000000000.0)
		tmp = t_1;
	elseif (t_4 <= 0.002)
		tmp = (x - (t_2 / (t * z))) / (x - -1.0);
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_3)) / (x - -1.0);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x - N[(t$95$2 / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -500000000000.0], t$95$1, If[LessEqual[t$95$4, 0.002], N[(N[(x - N[(t$95$2 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e11 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 82.8

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 97.0%

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

   if -5e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-3

   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 x around 0

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

      1. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 2e-3 < (/.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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   7. 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 99.5

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

   8. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -500000000000:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.002:\\ \;\;\;\;\frac{x - \frac{x - z \cdot y}{t \cdot z}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ t_2 := t \cdot z - x\\ t_3 := \frac{x - \frac{x - z \cdot y}{t\_2}}{x - -1}\\ \mathbf{if}\;t\_3 \leq -500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.002:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x - -1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (/ z (- -1.0 x)) (- x (* t z))) y))
        (t_2 (- (* t z) x))
        (t_3 (/ (- x (/ (- x (* z y)) t_2)) (- x -1.0))))
   (if (<= t_3 -500000000000.0)
     t_1
     (if (<= t_3 0.002)
       (/ (- x (/ (- (/ x z) y) t)) 1.0)
       (if (<= t_3 2.0)
         (/ (- x (/ x t_2)) (- x -1.0))
         (if (<= t_3 INFINITY) t_1 (/ (+ (/ y t) x) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double tmp;
	if (t_3 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_3 <= 0.002) {
		tmp = (x - (((x / z) - y) / t)) / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double tmp;
	if (t_3 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_3 <= 0.002) {
		tmp = (x - (((x / z) - y) / t)) / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y
	t_2 = (t * z) - x
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0)
	tmp = 0
	if t_3 <= -500000000000.0:
		tmp = t_1
	elif t_3 <= 0.002:
		tmp = (x - (((x / z) - y) / t)) / 1.0
	elif t_3 <= 2.0:
		tmp = (x - (x / t_2)) / (x - -1.0)
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z / Float64(-1.0 - x)) / Float64(x - Float64(t * z))) * y)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / t_2)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_3 <= -500000000000.0)
		tmp = t_1;
	elseif (t_3 <= 0.002)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / 1.0);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x - -1.0));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	t_2 = (t * z) - x;
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	tmp = 0.0;
	if (t_3 <= -500000000000.0)
		tmp = t_1;
	elseif (t_3 <= 0.002)
		tmp = (x - (((x / z) - y) / t)) / 1.0;
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_2)) / (x - -1.0);
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -500000000000.0], t$95$1, If[LessEqual[t$95$3, 0.002], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e11 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 82.8

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 97.0%

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

   if -5e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-3

   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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in t around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. sub-neg N/A

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

       5. mul-1-neg N/A

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

       6. remove-double-neg N/A

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

       7. lower-/.f64 N/A

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

       8. +-commutative N/A

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

       9. mul-1-neg N/A

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

       10. unsub-neg N/A

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

       11. lower--.f64 N/A

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

       12. lower-/.f64 98.3

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

   11. Applied rewrites 98.3%

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

   if 2e-3 < (/.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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   7. 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 99.5

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

   8. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -500000000000:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.002:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ t_2 := x - z \cdot y\\ t_3 := t \cdot z - x\\ t_4 := \frac{x - \frac{t\_2}{t\_3}}{x - -1}\\ \mathbf{if}\;t\_4 \leq -500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.002:\\ \;\;\;\;\frac{x - \frac{t\_2}{t \cdot z}}{1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_3}}{x - -1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (/ z (- -1.0 x)) (- x (* t z))) y))
        (t_2 (- x (* z y)))
        (t_3 (- (* t z) x))
        (t_4 (/ (- x (/ t_2 t_3)) (- x -1.0))))
   (if (<= t_4 -500000000000.0)
     t_1
     (if (<= t_4 0.002)
       (/ (- x (/ t_2 (* t z))) 1.0)
       (if (<= t_4 2.0)
         (/ (- x (/ x t_3)) (- x -1.0))
         (if (<= t_4 INFINITY) t_1 (/ (+ (/ y t) x) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = x - (z * y);
	double t_3 = (t * z) - x;
	double t_4 = (x - (t_2 / t_3)) / (x - -1.0);
	double tmp;
	if (t_4 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_4 <= 0.002) {
		tmp = (x - (t_2 / (t * z))) / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_3)) / (x - -1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = x - (z * y);
	double t_3 = (t * z) - x;
	double t_4 = (x - (t_2 / t_3)) / (x - -1.0);
	double tmp;
	if (t_4 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_4 <= 0.002) {
		tmp = (x - (t_2 / (t * z))) / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_3)) / (x - -1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y
	t_2 = x - (z * y)
	t_3 = (t * z) - x
	t_4 = (x - (t_2 / t_3)) / (x - -1.0)
	tmp = 0
	if t_4 <= -500000000000.0:
		tmp = t_1
	elif t_4 <= 0.002:
		tmp = (x - (t_2 / (t * z))) / 1.0
	elif t_4 <= 2.0:
		tmp = (x - (x / t_3)) / (x - -1.0)
	elif t_4 <= math.inf:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z / Float64(-1.0 - x)) / Float64(x - Float64(t * z))) * y)
	t_2 = Float64(x - Float64(z * y))
	t_3 = Float64(Float64(t * z) - x)
	t_4 = Float64(Float64(x - Float64(t_2 / t_3)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_4 <= -500000000000.0)
		tmp = t_1;
	elseif (t_4 <= 0.002)
		tmp = Float64(Float64(x - Float64(t_2 / Float64(t * z))) / 1.0);
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_3)) / Float64(x - -1.0));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	t_2 = x - (z * y);
	t_3 = (t * z) - x;
	t_4 = (x - (t_2 / t_3)) / (x - -1.0);
	tmp = 0.0;
	if (t_4 <= -500000000000.0)
		tmp = t_1;
	elseif (t_4 <= 0.002)
		tmp = (x - (t_2 / (t * z))) / 1.0;
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_3)) / (x - -1.0);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x - N[(t$95$2 / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -500000000000.0], t$95$1, If[LessEqual[t$95$4, 0.002], N[(N[(x - N[(t$95$2 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e11 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 82.8

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 97.0%

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

   if -5e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-3

   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 x around 0

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

      1. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

   if 2e-3 < (/.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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   7. 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 99.5

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

   8. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -500000000000:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.002:\\ \;\;\;\;\frac{x - \frac{x - z \cdot y}{t \cdot z}}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ t_2 := \frac{\frac{y}{t} + x}{x - -1}\\ t_3 := t \cdot z - x\\ t_4 := \frac{x - \frac{x - z \cdot y}{t\_3}}{x - -1}\\ \mathbf{if}\;t\_4 \leq -500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.002:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_3}}{x - -1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (/ z (- -1.0 x)) (- x (* t z))) y))
        (t_2 (/ (+ (/ y t) x) (- x -1.0)))
        (t_3 (- (* t z) x))
        (t_4 (/ (- x (/ (- x (* z y)) t_3)) (- x -1.0))))
   (if (<= t_4 -500000000000.0)
     t_1
     (if (<= t_4 0.002)
       t_2
       (if (<= t_4 2.0)
         (/ (- x (/ x t_3)) (- x -1.0))
         (if (<= t_4 INFINITY) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = ((y / t) + x) / (x - -1.0);
	double t_3 = (t * z) - x;
	double t_4 = (x - ((x - (z * y)) / t_3)) / (x - -1.0);
	double tmp;
	if (t_4 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_4 <= 0.002) {
		tmp = t_2;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_3)) / (x - -1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	double t_2 = ((y / t) + x) / (x - -1.0);
	double t_3 = (t * z) - x;
	double t_4 = (x - ((x - (z * y)) / t_3)) / (x - -1.0);
	double tmp;
	if (t_4 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_4 <= 0.002) {
		tmp = t_2;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_3)) / (x - -1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y
	t_2 = ((y / t) + x) / (x - -1.0)
	t_3 = (t * z) - x
	t_4 = (x - ((x - (z * y)) / t_3)) / (x - -1.0)
	tmp = 0
	if t_4 <= -500000000000.0:
		tmp = t_1
	elif t_4 <= 0.002:
		tmp = t_2
	elif t_4 <= 2.0:
		tmp = (x - (x / t_3)) / (x - -1.0)
	elif t_4 <= math.inf:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z / Float64(-1.0 - x)) / Float64(x - Float64(t * z))) * y)
	t_2 = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0))
	t_3 = Float64(Float64(t * z) - x)
	t_4 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / t_3)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_4 <= -500000000000.0)
		tmp = t_1;
	elseif (t_4 <= 0.002)
		tmp = t_2;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_3)) / Float64(x - -1.0));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((z / (-1.0 - x)) / (x - (t * z))) * y;
	t_2 = ((y / t) + x) / (x - -1.0);
	t_3 = (t * z) - x;
	t_4 = (x - ((x - (z * y)) / t_3)) / (x - -1.0);
	tmp = 0.0;
	if (t_4 <= -500000000000.0)
		tmp = t_1;
	elseif (t_4 <= 0.002)
		tmp = t_2;
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_3)) / (x - -1.0);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -500000000000.0], t$95$1, If[LessEqual[t$95$4, 0.002], t$95$2, If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, t$95$2]]]]]]]]

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))) < -5e11 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 82.8

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 97.0%

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

   if -5e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-3 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 83.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if 2e-3 < (/.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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   7. 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 99.5

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

   8. Applied rewrites 99.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -500000000000:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.002:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{-1 - x}}{x - t \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x - -1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{x - \frac{x - z \cdot y}{t\_2}}{x - -1}\\ t_4 := \left(-1 - x\right) \cdot \left(x - t \cdot z\right)\\ \mathbf{if}\;t\_3 \leq -500000000000:\\ \;\;\;\;\frac{z}{t\_4} \cdot y\\ \mathbf{elif}\;t\_3 \leq 0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x - -1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{z \cdot y}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (- x -1.0)))
        (t_2 (- (* t z) x))
        (t_3 (/ (- x (/ (- x (* z y)) t_2)) (- x -1.0)))
        (t_4 (* (- -1.0 x) (- x (* t z)))))
   (if (<= t_3 -500000000000.0)
     (* (/ z t_4) y)
     (if (<= t_3 0.002)
       t_1
       (if (<= t_3 2.0)
         (/ (- x (/ x t_2)) (- x -1.0))
         (if (<= t_3 5e+293) (/ (* z y) t_4) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double t_4 = (-1.0 - x) * (x - (t * z));
	double tmp;
	if (t_3 <= -500000000000.0) {
		tmp = (z / t_4) * y;
	} else if (t_3 <= 0.002) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_3 <= 5e+293) {
		tmp = (z * y) / 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 = ((y / t) + x) / (x - (-1.0d0))
    t_2 = (t * z) - x
    t_3 = (x - ((x - (z * y)) / t_2)) / (x - (-1.0d0))
    t_4 = ((-1.0d0) - x) * (x - (t * z))
    if (t_3 <= (-500000000000.0d0)) then
        tmp = (z / t_4) * y
    else if (t_3 <= 0.002d0) then
        tmp = t_1
    else if (t_3 <= 2.0d0) then
        tmp = (x - (x / t_2)) / (x - (-1.0d0))
    else if (t_3 <= 5d+293) then
        tmp = (z * y) / 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 = ((y / t) + x) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	double t_4 = (-1.0 - x) * (x - (t * z));
	double tmp;
	if (t_3 <= -500000000000.0) {
		tmp = (z / t_4) * y;
	} else if (t_3 <= 0.002) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_3 <= 5e+293) {
		tmp = (z * y) / t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x - -1.0)
	t_2 = (t * z) - x
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0)
	t_4 = (-1.0 - x) * (x - (t * z))
	tmp = 0
	if t_3 <= -500000000000.0:
		tmp = (z / t_4) * y
	elif t_3 <= 0.002:
		tmp = t_1
	elif t_3 <= 2.0:
		tmp = (x - (x / t_2)) / (x - -1.0)
	elif t_3 <= 5e+293:
		tmp = (z * y) / t_4
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / t_2)) / Float64(x - -1.0))
	t_4 = Float64(Float64(-1.0 - x) * Float64(x - Float64(t * z)))
	tmp = 0.0
	if (t_3 <= -500000000000.0)
		tmp = Float64(Float64(z / t_4) * y);
	elseif (t_3 <= 0.002)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x - -1.0));
	elseif (t_3 <= 5e+293)
		tmp = Float64(Float64(z * y) / t_4);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x - -1.0);
	t_2 = (t * z) - x;
	t_3 = (x - ((x - (z * y)) / t_2)) / (x - -1.0);
	t_4 = (-1.0 - x) * (x - (t * z));
	tmp = 0.0;
	if (t_3 <= -500000000000.0)
		tmp = (z / t_4) * y;
	elseif (t_3 <= 0.002)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_2)) / (x - -1.0);
	elseif (t_3 <= 5e+293)
		tmp = (z * y) / t_4;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 - x), $MachinePrecision] * N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -500000000000.0], N[(N[(z / t$95$4), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$3, 0.002], t$95$1, If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+293], N[(N[(z * y), $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))) < -5e11

   1. Initial program 73.6%

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

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.2

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 93.1%

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

   if -5e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e-3 or 5.00000000000000033e293 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if 2e-3 < (/.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 z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   7. 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 99.5

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

   8. Applied rewrites 99.5%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000033e293

   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. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 99.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -500000000000:\\ \;\;\;\;\frac{z}{\left(-1 - x\right) \cdot \left(x - t \cdot z\right)} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.002:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{z \cdot y}{\left(-1 - x\right) \cdot \left(x - t \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x - -1}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ t_3 := \left(-1 - x\right) \cdot \left(x - t \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -500000000000:\\ \;\;\;\;\frac{z}{t\_3} \cdot y\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{z \cdot y}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (- x -1.0)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)))
        (t_3 (* (- -1.0 x) (- x (* t z)))))
   (if (<= t_2 -500000000000.0)
     (* (/ z t_3) y)
     (if (<= t_2 0.2)
       t_1
       (if (<= t_2 2.0) 1.0 (if (<= t_2 5e+293) (/ (* z y) t_3) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_3 = (-1.0 - x) * (x - (t * z));
	double tmp;
	if (t_2 <= -500000000000.0) {
		tmp = (z / t_3) * y;
	} else if (t_2 <= 0.2) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+293) {
		tmp = (z * y) / 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) :: tmp
    t_1 = ((y / t) + x) / (x - (-1.0d0))
    t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    t_3 = ((-1.0d0) - x) * (x - (t * z))
    if (t_2 <= (-500000000000.0d0)) then
        tmp = (z / t_3) * y
    else if (t_2 <= 0.2d0) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else if (t_2 <= 5d+293) then
        tmp = (z * y) / 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 = ((y / t) + x) / (x - -1.0);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_3 = (-1.0 - x) * (x - (t * z));
	double tmp;
	if (t_2 <= -500000000000.0) {
		tmp = (z / t_3) * y;
	} else if (t_2 <= 0.2) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+293) {
		tmp = (z * y) / t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x - -1.0)
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	t_3 = (-1.0 - x) * (x - (t * z))
	tmp = 0
	if t_2 <= -500000000000.0:
		tmp = (z / t_3) * y
	elif t_2 <= 0.2:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = 1.0
	elif t_2 <= 5e+293:
		tmp = (z * y) / t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	t_3 = Float64(Float64(-1.0 - x) * Float64(x - Float64(t * z)))
	tmp = 0.0
	if (t_2 <= -500000000000.0)
		tmp = Float64(Float64(z / t_3) * y);
	elseif (t_2 <= 0.2)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+293)
		tmp = Float64(Float64(z * y) / t_3);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x - -1.0);
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	t_3 = (-1.0 - x) * (x - (t * z));
	tmp = 0.0;
	if (t_2 <= -500000000000.0)
		tmp = (z / t_3) * y;
	elseif (t_2 <= 0.2)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+293)
		tmp = (z * y) / t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.0 - x), $MachinePrecision] * N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500000000000.0], N[(N[(z / t$95$3), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, 0.2], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, 5e+293], N[(N[(z * y), $MachinePrecision] / t$95$3), $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))) < -5e11

   1. Initial program 73.6%

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

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 81.2

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 93.1%

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

   if -5e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001 or 5.00000000000000033e293 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 88.1

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

   5. Applied rewrites 88.1%

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

   if 0.20000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.6%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000033e293

   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. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 99.2%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -500000000000:\\ \;\;\;\;\frac{z}{\left(-1 - x\right) \cdot \left(x - t \cdot z\right)} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.2:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{z \cdot y}{\left(-1 - x\right) \cdot \left(x - t \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{\left(-1 - x\right) \cdot \left(x - t \cdot z\right)} \cdot y\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ t_3 := \frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{if}\;t\_2 \leq -500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (* (- -1.0 x) (- x (* t z)))) y))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0)))
        (t_3 (/ (+ (/ y t) x) (- x -1.0))))
   (if (<= t_2 -500000000000.0)
     t_1
     (if (<= t_2 0.2)
       t_3
       (if (<= t_2 2.0) 1.0 (if (<= t_2 5e+293) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / ((-1.0 - x) * (x - (t * z)))) * y;
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_3 = ((y / t) + x) / (x - -1.0);
	double tmp;
	if (t_2 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.2) {
		tmp = t_3;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+293) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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 / (((-1.0d0) - x) * (x - (t * z)))) * y
    t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    t_3 = ((y / t) + x) / (x - (-1.0d0))
    if (t_2 <= (-500000000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 0.2d0) then
        tmp = t_3
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else if (t_2 <= 5d+293) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / ((-1.0 - x) * (x - (t * z)))) * y;
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double t_3 = ((y / t) + x) / (x - -1.0);
	double tmp;
	if (t_2 <= -500000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.2) {
		tmp = t_3;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+293) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / ((-1.0 - x) * (x - (t * z)))) * y
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	t_3 = ((y / t) + x) / (x - -1.0)
	tmp = 0
	if t_2 <= -500000000000.0:
		tmp = t_1
	elif t_2 <= 0.2:
		tmp = t_3
	elif t_2 <= 2.0:
		tmp = 1.0
	elif t_2 <= 5e+293:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / Float64(Float64(-1.0 - x) * Float64(x - Float64(t * z)))) * y)
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	t_3 = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= -500000000000.0)
		tmp = t_1;
	elseif (t_2 <= 0.2)
		tmp = t_3;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+293)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / ((-1.0 - x) * (x - (t * z)))) * y;
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	t_3 = ((y / t) + x) / (x - -1.0);
	tmp = 0.0;
	if (t_2 <= -500000000000.0)
		tmp = t_1;
	elseif (t_2 <= 0.2)
		tmp = t_3;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+293)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(N[(-1.0 - x), $MachinePrecision] * N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500000000000.0], t$95$1, If[LessEqual[t$95$2, 0.2], t$95$3, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, 5e+293], t$95$1, t$95$3]]]]]]]

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))) < -5e11 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000033e293

   1. Initial program 82.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 79.8

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

   5. Applied rewrites 79.8%

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

   7. Applied rewrites 95.1%

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

   if -5e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001 or 5.00000000000000033e293 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 88.1

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

   5. Applied rewrites 88.1%

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

   if 0.20000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.6%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -500000000000:\\ \;\;\;\;\frac{z}{\left(-1 - x\right) \cdot \left(x - t \cdot z\right)} \cdot y\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.2:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{z}{\left(-1 - x\right) \cdot \left(x - t \cdot z\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+139}:\\ \;\;\;\;\frac{z \cdot y}{\left(-1 - x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 0.2)
     (/ (+ (/ y t) x) 1.0)
     (if (<= t_1 2.0)
       1.0
       (if (<= t_1 4e+139)
         (/ (* z y) (* (- -1.0 x) x))
         (/ y (* (- x -1.0) t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 0.2) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= 4e+139) {
		tmp = (z * y) / ((-1.0 - x) * x);
	} else {
		tmp = y / ((x - -1.0) * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= 0.2d0) then
        tmp = ((y / t) + x) / 1.0d0
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else if (t_1 <= 4d+139) then
        tmp = (z * y) / (((-1.0d0) - x) * x)
    else
        tmp = y / ((x - (-1.0d0)) * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 0.2) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= 4e+139) {
		tmp = (z * y) / ((-1.0 - x) * x);
	} else {
		tmp = y / ((x - -1.0) * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= 0.2:
		tmp = ((y / t) + x) / 1.0
	elif t_1 <= 2.0:
		tmp = 1.0
	elif t_1 <= 4e+139:
		tmp = (z * y) / ((-1.0 - x) * x)
	else:
		tmp = y / ((x - -1.0) * t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 0.2)
		tmp = Float64(Float64(Float64(y / t) + x) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= 4e+139)
		tmp = Float64(Float64(z * y) / Float64(Float64(-1.0 - x) * x));
	else
		tmp = Float64(y / Float64(Float64(x - -1.0) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= 0.2)
		tmp = ((y / t) + x) / 1.0;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= 4e+139)
		tmp = (z * y) / ((-1.0 - x) * x);
	else
		tmp = y / ((x - -1.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.2], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, If[LessEqual[t$95$1, 4e+139], N[(N[(z * y), $MachinePrecision] / N[(N[(-1.0 - x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(x - -1.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 90.3%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.8%

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

   if 0.20000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.6%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.00000000000000013e139

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.4%

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

   if 4.00000000000000013e139 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 46.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 61.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.2:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 4 \cdot 10^{+139}:\\ \;\;\;\;\frac{z \cdot y}{\left(-1 - x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(x - -1\right) \cdot t}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_2 \leq -0.4:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (- x -1.0) t)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_2 -0.4)
     t_1
     (if (<= t_2 0.2) (/ x (- x -1.0)) (if (<= t_2 2e+24) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / ((x - -1.0) * t);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= -0.4) {
		tmp = t_1;
	} else if (t_2 <= 0.2) {
		tmp = x / (x - -1.0);
	} else if (t_2 <= 2e+24) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y / ((x - (-1.0d0)) * t)
    t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_2 <= (-0.4d0)) then
        tmp = t_1
    else if (t_2 <= 0.2d0) then
        tmp = x / (x - (-1.0d0))
    else if (t_2 <= 2d+24) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / ((x - -1.0) * t);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= -0.4) {
		tmp = t_1;
	} else if (t_2 <= 0.2) {
		tmp = x / (x - -1.0);
	} else if (t_2 <= 2e+24) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / ((x - -1.0) * t)
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_2 <= -0.4:
		tmp = t_1
	elif t_2 <= 0.2:
		tmp = x / (x - -1.0)
	elif t_2 <= 2e+24:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(x - -1.0) * t))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= -0.4)
		tmp = t_1;
	elseif (t_2 <= 0.2)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_2 <= 2e+24)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / ((x - -1.0) * t);
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_2 <= -0.4)
		tmp = t_1;
	elseif (t_2 <= 0.2)
		tmp = x / (x - -1.0);
	elseif (t_2 <= 2e+24)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(x - -1.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.4], t$95$1, If[LessEqual[t$95$2, 0.2], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+24], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -0.40000000000000002 or 2e24 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 56.2%

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

   if -0.40000000000000002 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001

   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 t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 58.1

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

   5. Applied rewrites 58.1%

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

   if 0.20000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e24

   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 95.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -0.4:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq -0.4:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 -0.4)
     (/ y t)
     (if (<= t_1 0.2) (/ x (- x -1.0)) (if (<= t_1 2e+24) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -0.4) {
		tmp = y / t;
	} else if (t_1 <= 0.2) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 2e+24) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= (-0.4d0)) then
        tmp = y / t
    else if (t_1 <= 0.2d0) then
        tmp = x / (x - (-1.0d0))
    else if (t_1 <= 2d+24) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -0.4) {
		tmp = y / t;
	} else if (t_1 <= 0.2) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 2e+24) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= -0.4:
		tmp = y / t
	elif t_1 <= 0.2:
		tmp = x / (x - -1.0)
	elif t_1 <= 2e+24:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= -0.4)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.2)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_1 <= 2e+24)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= -0.4)
		tmp = y / t;
	elseif (t_1 <= 0.2)
		tmp = x / (x - -1.0);
	elseif (t_1 <= 2e+24)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.4], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.2], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+24], 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))) < -0.40000000000000002 or 2e24 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -0.40000000000000002 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001

   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 t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 58.1

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

   5. Applied rewrites 58.1%

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

   if 0.20000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e24

   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 95.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -0.4:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(-1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 -1e-44)
     (/ y t)
     (if (<= t_1 2e-40)
       (* (fma -1.0 x 1.0) x)
       (if (<= t_1 2e+24) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -1e-44) {
		tmp = y / t;
	} else if (t_1 <= 2e-40) {
		tmp = fma(-1.0, x, 1.0) * x;
	} else if (t_1 <= 2e+24) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= -1e-44)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-40)
		tmp = Float64(fma(-1.0, x, 1.0) * x);
	elseif (t_1 <= 2e+24)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-44], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-40], N[(N[(-1.0 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2e+24], 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))) < -9.99999999999999953e-45 or 2e24 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 49.7

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

   5. Applied rewrites 49.7%

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

   if -9.99999999999999953e-45 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-40

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

      5. difference-of-squares N/A

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

      6. lift--.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. neg-sub0 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. lower--.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.8

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.8%

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

   if 1.9999999999999999e-40 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -1 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(-1, x, 1\right) \cdot x\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{y}{\frac{-1 - x}{z} \cdot \left(x - t \cdot z\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 -2e+125)
     (/ y (* (/ (- -1.0 x) z) (- x (* t z))))
     (if (<= t_1 5e+293) t_1 (/ (+ (/ y t) x) (- x -1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -2e+125) {
		tmp = y / (((-1.0 - x) / z) * (x - (t * z)));
	} else if (t_1 <= 5e+293) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= (-2d+125)) then
        tmp = y / ((((-1.0d0) - x) / z) * (x - (t * z)))
    else if (t_1 <= 5d+293) then
        tmp = t_1
    else
        tmp = ((y / t) + x) / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -2e+125) {
		tmp = y / (((-1.0 - x) / z) * (x - (t * z)));
	} else if (t_1 <= 5e+293) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= -2e+125:
		tmp = y / (((-1.0 - x) / z) * (x - (t * z)))
	elif t_1 <= 5e+293:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= -2e+125)
		tmp = Float64(y / Float64(Float64(Float64(-1.0 - x) / z) * Float64(x - Float64(t * z))));
	elseif (t_1 <= 5e+293)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= -2e+125)
		tmp = y / (((-1.0 - x) / z) * (x - (t * z)));
	elseif (t_1 <= 5e+293)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+125], N[(y / N[(N[(N[(-1.0 - x), $MachinePrecision] / z), $MachinePrecision] * N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+293], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.9999999999999998e125

   1. Initial program 62.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 94.1%

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

   if -1.9999999999999998e125 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000033e293

   1. Initial program 99.8%

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

   if 5.00000000000000033e293 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 31.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 95.0

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

   5. Applied rewrites 95.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{y}{\frac{-1 - x}{z} \cdot \left(x - t \cdot z\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x - -1}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_2 \leq 0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (- x -1.0)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_2 0.2) t_1 (if (<= t_2 1.0) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= 0.2) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((y / t) + x) / (x - (-1.0d0))
    t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_2 <= 0.2d0) then
        tmp = t_1
    else if (t_2 <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= 0.2) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x - -1.0)
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_2 <= 0.2:
		tmp = t_1
	elif t_2 <= 1.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= 0.2)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x - -1.0);
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_2 <= 0.2)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.2], t$95$1, If[LessEqual[t$95$2, 1.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.20000000000000001 or 1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if 0.20000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.2:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 0.2)
     (/ (+ (/ y t) x) 1.0)
     (if (<= t_1 2e+24) 1.0 (/ y (* (- x -1.0) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 0.2) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2e+24) {
		tmp = 1.0;
	} else {
		tmp = y / ((x - -1.0) * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= 0.2d0) then
        tmp = ((y / t) + x) / 1.0d0
    else if (t_1 <= 2d+24) then
        tmp = 1.0d0
    else
        tmp = y / ((x - (-1.0d0)) * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 0.2) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2e+24) {
		tmp = 1.0;
	} else {
		tmp = y / ((x - -1.0) * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= 0.2:
		tmp = ((y / t) + x) / 1.0
	elif t_1 <= 2e+24:
		tmp = 1.0
	else:
		tmp = y / ((x - -1.0) * t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 0.2)
		tmp = Float64(Float64(Float64(y / t) + x) / 1.0);
	elseif (t_1 <= 2e+24)
		tmp = 1.0;
	else
		tmp = Float64(y / Float64(Float64(x - -1.0) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= 0.2)
		tmp = ((y / t) + x) / 1.0;
	elseif (t_1 <= 2e+24)
		tmp = 1.0;
	else
		tmp = y / ((x - -1.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.2], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+24], 1.0, N[(y / N[(N[(x - -1.0), $MachinePrecision] * t), $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))) < 0.20000000000000001

   1. Initial program 90.3%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.8%

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

   if 0.20000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e24

   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 95.6%

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

   if 2e24 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 59.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 55.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.2:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 0.002) (/ y t) (if (<= t_1 2e+24) 1.0 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 0.002) {
		tmp = y / t;
	} else if (t_1 <= 2e+24) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= 0.002d0) then
        tmp = y / t
    else if (t_1 <= 2d+24) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 0.002) {
		tmp = y / t;
	} else if (t_1 <= 2e+24) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= 0.002:
		tmp = y / t
	elif t_1 <= 2e+24:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 0.002)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e+24)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= 0.002)
		tmp = y / t;
	elseif (t_1 <= 2e+24)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.002], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e+24], 1.0, N[(y / t), $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))) < 2e-3 or 2e24 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 41.4

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

   5. Applied rewrites 41.4%

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

   if 2e-3 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e24

   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 95.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 0.002:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.8% 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 91.1%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 53.0%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024331 
(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)))