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

Percentage Accurate: 89.7% → 94.9%

Time: 11.8s

Alternatives: 13

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.7% 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: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{x + \frac{\frac{1}{z \cdot t - x}}{\frac{1}{\mathsf{fma}\left(y, z, -x\right)}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)) 5e+289)
   (/ (+ x (/ (/ 1.0 (- (* z t) x)) (/ 1.0 (fma y z (- x))))) (+ x 1.0))
   (/ (+ x (/ y t)) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0)) <= 5e+289) {
		tmp = (x + ((1.0 / ((z * t) - x)) / (1.0 / fma(y, z, -x)))) / (x + 1.0);
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0)) <= 5e+289)
		tmp = Float64(Float64(x + Float64(Float64(1.0 / Float64(Float64(z * t) - x)) / Float64(1.0 / fma(y, z, Float64(-x))))) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 5e+289], N[(N[(x + N[(N[(1.0 / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000031e289

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-inv N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. frac-2neg N/A

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

      9. clear-num N/A

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

      10. frac-2neg N/A

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

      11. clear-num N/A

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

      12. flip-- N/A

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

      13. lift--.f64 N/A

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

      14. associate-*l/ N/A

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

   4. Applied rewrites 97.7%

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

   if 5.00000000000000031e289 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{x + \frac{\frac{1}{z \cdot t - x}}{\frac{1}{\mathsf{fma}\left(y, z, -x\right)}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ t_2 := x - z \cdot t\\ t_3 := \frac{x + \frac{x - y \cdot z}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -400000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.5:\\ \;\;\;\;\frac{\frac{\frac{x}{z} - y}{t} - x}{-1 - x}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) (* (- (* z t) x) (+ x 1.0))))
        (t_2 (- x (* z t)))
        (t_3 (/ (+ x (/ (- x (* y z)) t_2)) (+ x 1.0))))
   (if (<= t_3 -400000000.0)
     t_1
     (if (<= t_3 0.5)
       (/ (- (/ (- (/ x z) y) t) x) (- -1.0 x))
       (if (<= t_3 2.0)
         (/ (+ x (/ x t_2)) (+ x 1.0))
         (if (<= t_3 5e+289) t_1 (/ (+ x (/ y t)) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / (((z * t) - x) * (x + 1.0));
	double t_2 = x - (z * t);
	double t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -400000000.0) {
		tmp = t_1;
	} else if (t_3 <= 0.5) {
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x);
	} else if (t_3 <= 2.0) {
		tmp = (x + (x / t_2)) / (x + 1.0);
	} else if (t_3 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (y * z) / (((z * t) - x) * (x + 1.0d0))
    t_2 = x - (z * t)
    t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0d0)
    if (t_3 <= (-400000000.0d0)) then
        tmp = t_1
    else if (t_3 <= 0.5d0) then
        tmp = ((((x / z) - y) / t) - x) / ((-1.0d0) - x)
    else if (t_3 <= 2.0d0) then
        tmp = (x + (x / t_2)) / (x + 1.0d0)
    else if (t_3 <= 5d+289) then
        tmp = t_1
    else
        tmp = (x + (y / t)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / (((z * t) - x) * (x + 1.0));
	double t_2 = x - (z * t);
	double t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -400000000.0) {
		tmp = t_1;
	} else if (t_3 <= 0.5) {
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x);
	} else if (t_3 <= 2.0) {
		tmp = (x + (x / t_2)) / (x + 1.0);
	} else if (t_3 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) / (((z * t) - x) * (x + 1.0))
	t_2 = x - (z * t)
	t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -400000000.0:
		tmp = t_1
	elif t_3 <= 0.5:
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x)
	elif t_3 <= 2.0:
		tmp = (x + (x / t_2)) / (x + 1.0)
	elif t_3 <= 5e+289:
		tmp = t_1
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / Float64(Float64(Float64(z * t) - x) * Float64(x + 1.0)))
	t_2 = Float64(x - Float64(z * t))
	t_3 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -400000000.0)
		tmp = t_1;
	elseif (t_3 <= 0.5)
		tmp = Float64(Float64(Float64(Float64(Float64(x / z) - y) / t) - x) / Float64(-1.0 - x));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x + Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_3 <= 5e+289)
		tmp = t_1;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) / (((z * t) - x) * (x + 1.0));
	t_2 = x - (z * t);
	t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -400000000.0)
		tmp = t_1;
	elseif (t_3 <= 0.5)
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x);
	elseif (t_3 <= 2.0)
		tmp = (x + (x / t_2)) / (x + 1.0);
	elseif (t_3 <= 5e+289)
		tmp = t_1;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / N[(N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -400000000.0], t$95$1, If[LessEqual[t$95$3, 0.5], N[(N[(N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $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, 5e+289], t$95$1, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4e8 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000031e289

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if -4e8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 95.0%

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

   3. Taylor expanded in t around -inf

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   if 0.5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.00000000000000031e289 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -400000000:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 0.5:\\ \;\;\;\;\frac{\frac{\frac{x}{z} - y}{t} - x}{-1 - x}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ t_2 := x - z \cdot t\\ t_3 := \frac{x + \frac{x - y \cdot z}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -400000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.5:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(y, z, -x\right)}{z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) (* (- (* z t) x) (+ x 1.0))))
        (t_2 (- x (* z t)))
        (t_3 (/ (+ x (/ (- x (* y z)) t_2)) (+ x 1.0))))
   (if (<= t_3 -400000000.0)
     t_1
     (if (<= t_3 0.5)
       (/ (+ x (/ (fma y z (- x)) (* z t))) (+ x 1.0))
       (if (<= t_3 2.0)
         (/ (+ x (/ x t_2)) (+ x 1.0))
         (if (<= t_3 5e+289) t_1 (/ (+ x (/ y t)) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / (((z * t) - x) * (x + 1.0));
	double t_2 = x - (z * t);
	double t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -400000000.0) {
		tmp = t_1;
	} else if (t_3 <= 0.5) {
		tmp = (x + (fma(y, z, -x) / (z * t))) / (x + 1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x + (x / t_2)) / (x + 1.0);
	} else if (t_3 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / Float64(Float64(Float64(z * t) - x) * Float64(x + 1.0)))
	t_2 = Float64(x - Float64(z * t))
	t_3 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -400000000.0)
		tmp = t_1;
	elseif (t_3 <= 0.5)
		tmp = Float64(Float64(x + Float64(fma(y, z, Float64(-x)) / Float64(z * 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 <= 5e+289)
		tmp = t_1;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / N[(N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -400000000.0], t$95$1, If[LessEqual[t$95$3, 0.5], N[(N[(x + N[(N[(y * z + (-x)), $MachinePrecision] / N[(z * t), $MachinePrecision]), $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, 5e+289], t$95$1, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4e8 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000031e289

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if -4e8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 95.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 0.5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.00000000000000031e289 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -400000000:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 0.5:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(y, z, -x\right)}{z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ t_3 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -400000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.999999999950755:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) (* (- (* z t) x) (+ x 1.0))))
        (t_2 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)))
        (t_3 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t_2 -400000000.0)
     t_1
     (if (<= t_2 0.999999999950755)
       t_3
       (if (<= t_2 2.0) 1.0 (if (<= t_2 5e+289) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / (((z * t) - x) * (x + 1.0));
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double t_3 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t_2 <= -400000000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.999999999950755) {
		tmp = t_3;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+289) {
		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 = (y * z) / (((z * t) - x) * (x + 1.0d0))
    t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0d0)
    t_3 = (x + (y / t)) / (x + 1.0d0)
    if (t_2 <= (-400000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 0.999999999950755d0) then
        tmp = t_3
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else if (t_2 <= 5d+289) 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 = (y * z) / (((z * t) - x) * (x + 1.0));
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double t_3 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t_2 <= -400000000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.999999999950755) {
		tmp = t_3;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) / (((z * t) - x) * (x + 1.0))
	t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0)
	t_3 = (x + (y / t)) / (x + 1.0)
	tmp = 0
	if t_2 <= -400000000.0:
		tmp = t_1
	elif t_2 <= 0.999999999950755:
		tmp = t_3
	elif t_2 <= 2.0:
		tmp = 1.0
	elif t_2 <= 5e+289:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / Float64(Float64(Float64(z * t) - x) * Float64(x + 1.0)))
	t_2 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	t_3 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -400000000.0)
		tmp = t_1;
	elseif (t_2 <= 0.999999999950755)
		tmp = t_3;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+289)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) / (((z * t) - x) * (x + 1.0));
	t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	t_3 = (x + (y / t)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -400000000.0)
		tmp = t_1;
	elseif (t_2 <= 0.999999999950755)
		tmp = t_3;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+289)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / N[(N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -400000000.0], t$95$1, If[LessEqual[t$95$2, 0.999999999950755], t$95$3, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, 5e+289], 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))) < -4e8 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000031e289

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if -4e8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999999995075495 or 5.00000000000000031e289 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if 0.99999999995075495 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -400000000:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 0.999999999950755:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ t_2 := x - z \cdot t\\ t_3 := \frac{x + \frac{x - y \cdot z}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) (* (- (* z t) x) (+ x 1.0))))
        (t_2 (- x (* z t)))
        (t_3 (/ (+ x (/ (- x (* y z)) t_2)) (+ x 1.0))))
   (if (<= t_3 -1.0)
     t_1
     (if (<= t_3 2.0)
       (/ (+ x (/ x t_2)) (+ x 1.0))
       (if (<= t_3 5e+289) t_1 (/ (+ x (/ y t)) (+ x 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / (((z * t) - x) * (x + 1.0));
	double t_2 = x - (z * t);
	double t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1.0) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x + (x / t_2)) / (x + 1.0);
	} else if (t_3 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (y * z) / (((z * t) - x) * (x + 1.0d0))
    t_2 = x - (z * t)
    t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0d0)
    if (t_3 <= (-1.0d0)) then
        tmp = t_1
    else if (t_3 <= 2.0d0) then
        tmp = (x + (x / t_2)) / (x + 1.0d0)
    else if (t_3 <= 5d+289) then
        tmp = t_1
    else
        tmp = (x + (y / t)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / (((z * t) - x) * (x + 1.0));
	double t_2 = x - (z * t);
	double t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1.0) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x + (x / t_2)) / (x + 1.0);
	} else if (t_3 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) / (((z * t) - x) * (x + 1.0))
	t_2 = x - (z * t)
	t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -1.0:
		tmp = t_1
	elif t_3 <= 2.0:
		tmp = (x + (x / t_2)) / (x + 1.0)
	elif t_3 <= 5e+289:
		tmp = t_1
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / Float64(Float64(Float64(z * t) - x) * Float64(x + 1.0)))
	t_2 = Float64(x - Float64(z * t))
	t_3 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x + Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_3 <= 5e+289)
		tmp = t_1;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) / (((z * t) - x) * (x + 1.0));
	t_2 = x - (z * t);
	t_3 = (x + ((x - (y * z)) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -1.0)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = (x + (x / t_2)) / (x + 1.0);
	elseif (t_3 <= 5e+289)
		tmp = t_1;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / N[(N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], 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+289], t$95$1, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000031e289

   1. Initial program 95.0%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 93.2

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

   5. Applied rewrites 93.2%

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

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

   1. Initial program 98.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if 5.00000000000000031e289 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -1:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ \mathbf{elif}\;t\_2 \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_2 -1.0)
     t_1
     (if (<= t_2 5e-15)
       (* x (+ 1.0 (/ -1.0 (* z t))))
       (if (<= t_2 50.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= 5e-15) {
		tmp = x * (1.0 + (-1.0 / (z * t)));
	} else if (t_2 <= 50.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= 5e-15)
		tmp = Float64(x * Float64(1.0 + Float64(-1.0 / Float64(z * t))));
	elseif (t_2 <= 50.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, 5e-15], N[(x * N[(1.0 + N[(-1.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 50.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1 or 50 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.0%

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

   3. Taylor expanded in t around -inf

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 69.8

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

   11. Applied rewrites 69.8%

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

   if -1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999999e-15

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 74.1

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 72.1

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

   8. Applied rewrites 72.1%

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

   if 4.99999999999999999e-15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 50

   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 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -1:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.999999999950755:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_2 -1.0)
     t_1
     (if (<= t_2 0.999999999950755)
       (/ x (+ x 1.0))
       (if (<= t_2 50.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= 0.999999999950755) {
		tmp = x / (x + 1.0);
	} else if (t_2 <= 50.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= 0.999999999950755)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_2 <= 50.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, 0.999999999950755], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 50.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1 or 50 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.0%

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

   3. Taylor expanded in t around -inf

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 69.8

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

   11. Applied rewrites 69.8%

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

   if -1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999999995075495

   1. Initial program 95.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 54.5

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

   5. Applied rewrites 54.5%

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

   if 0.99999999995075495 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 50

   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 98.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -1:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 0.999999999950755:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.999999999950755:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_1 -1.0)
     (/ y t)
     (if (<= t_1 0.999999999950755)
       (/ x (+ x 1.0))
       (if (<= t_1 50.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = y / t;
	} else if (t_1 <= 0.999999999950755) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 50.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0d0)
    if (t_1 <= (-1.0d0)) then
        tmp = y / t
    else if (t_1 <= 0.999999999950755d0) then
        tmp = x / (x + 1.0d0)
    else if (t_1 <= 50.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = y / t;
	} else if (t_1 <= 0.999999999950755) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 50.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0)
	tmp = 0
	if t_1 <= -1.0:
		tmp = y / t
	elif t_1 <= 0.999999999950755:
		tmp = x / (x + 1.0)
	elif t_1 <= 50.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.999999999950755)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_1 <= 50.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = y / t;
	elseif (t_1 <= 0.999999999950755)
		tmp = x / (x + 1.0);
	elseif (t_1 <= 50.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.999999999950755], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1 or 50 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 60.4

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

   5. Applied rewrites 60.4%

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

   if -1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999999995075495

   1. Initial program 95.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 54.5

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

   5. Applied rewrites 54.5%

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

   if 0.99999999995075495 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 50

   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 98.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -1:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 0.999999999950755:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_1 -1.0)
     (/ y t)
     (if (<= t_1 2e-29) (fma x (- x) x) (if (<= t_1 50.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = y / t;
	} else if (t_1 <= 2e-29) {
		tmp = fma(x, -x, x);
	} else if (t_1 <= 50.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-29)
		tmp = fma(x, Float64(-x), x);
	elseif (t_1 <= 50.0)
		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[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-29], N[(x * (-x) + x), $MachinePrecision], If[LessEqual[t$95$1, 50.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1 or 50 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 60.4

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

   5. Applied rewrites 60.4%

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

   if -1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999989e-29

   1. Initial program 94.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 55.1

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

   8. Applied rewrites 55.1%

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

   if 1.99999999999999989e-29 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 50

   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 94.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -1:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 0.999999999950755:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_2 0.999999999950755) t_1 (if (<= t_2 50.0) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.999999999950755) {
		tmp = t_1;
	} else if (t_2 <= 50.0) {
		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 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0d0)
    if (t_2 <= 0.999999999950755d0) then
        tmp = t_1
    else if (t_2 <= 50.0d0) 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 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.999999999950755) {
		tmp = t_1;
	} else if (t_2 <= 50.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0)
	tmp = 0
	if t_2 <= 0.999999999950755:
		tmp = t_1
	elif t_2 <= 50.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 0.999999999950755)
		tmp = t_1;
	elseif (t_2 <= 50.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 0.999999999950755)
		tmp = t_1;
	elseif (t_2 <= 50.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.999999999950755], t$95$1, If[LessEqual[t$95$2, 50.0], 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))) < 0.99999999995075495 or 50 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 81.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if 0.99999999995075495 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 50

   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 98.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 0.999999999950755:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_1 5e+289) t_1 (/ (+ x (/ y t)) (+ x 1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0d0)
    if (t_1 <= 5d+289) then
        tmp = t_1
    else
        tmp = (x + (y / t)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0)
	tmp = 0
	if t_1 <= 5e+289:
		tmp = t_1
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= 5e+289)
		tmp = t_1;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= 5e+289)
		tmp = t_1;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+289], t$95$1, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000031e289

   1. Initial program 97.7%

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

   if 5.00000000000000031e289 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)) 2e-29)
   (fma x (- x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0)) <= 2e-29) {
		tmp = fma(x, -x, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0)) <= 2e-29)
		tmp = fma(x, Float64(-x), x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 2e-29], N[(x * (-x) + x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999989e-29

   1. Initial program 93.4%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 27.4

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 28.1

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

   8. Applied rewrites 28.1%

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

   if 1.99999999999999989e-29 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 74.0%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.7% 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 90.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 52.2%

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

Developer Target 1: 99.6% 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 2024216 
(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)))