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

Percentage Accurate: 89.9% → 97.3%

Time: 10.3s

Alternatives: 20

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\frac{1 + x}{z} \cdot \mathsf{fma}\left(z, t, -x\right)}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_2 \leq -4:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-11}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1 + x}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (/ (+ 1.0 x) z) (fma z t (- x)))))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_2 -4.0)
     t_1
     (if (<= t_2 1e-11)
       (/ (- x (/ (- (/ x z) y) t)) (+ 1.0 x))
       (if (<= t_2 2.0)
         (/ (- x (/ x (fma t z (- x)))) (+ 1.0 x))
         (if (<= t_2 INFINITY) t_1 (/ (+ (/ y t) x) (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (((1.0 + x) / z) * fma(z, t, -x));
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= -4.0) {
		tmp = t_1;
	} else if (t_2 <= 1e-11) {
		tmp = (x - (((x / z) - y) / t)) / (1.0 + x);
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / fma(t, z, -x))) / (1.0 + x);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(Float64(1.0 + x) / z) * fma(z, t, Float64(-x))))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= -4.0)
		tmp = t_1;
	elseif (t_2 <= 1e-11)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(1.0 + x));
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(1.0 + x));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(N[(1.0 + x), $MachinePrecision] / z), $MachinePrecision] * N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4.0], t$95$1, If[LessEqual[t$95$2, 1e-11], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 78.4%

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

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

      12. lower-+.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 96.3%

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

   if -4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999939e-12

   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

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity 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 99.2

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

   5. Applied rewrites 99.2%

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

   if 9.99999999999999939e-12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -4:\\ \;\;\;\;\frac{y}{\frac{1 + x}{z} \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 10^{-11}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq \infty:\\ \;\;\;\;\frac{y}{\frac{1 + x}{z} \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ t_2 := \frac{y}{t} + x\\ t_3 := \frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;t\_1 \leq -4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{t\_2}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x)))
        (t_2 (+ (/ y t) x))
        (t_3 (/ (* z y) (* (fma z t (- x)) (+ 1.0 x)))))
   (if (<= t_1 (- INFINITY))
     (/ y (* (+ 1.0 x) t))
     (if (<= t_1 -4.0)
       t_3
       (if (<= t_1 -5e-168)
         (/ (/ (- y (/ x z)) t) 1.0)
         (if (<= t_1 5e-46)
           (/ t_2 1.0)
           (if (<= t_1 2.0)
             (/ (- x (/ x (fma t z (- x)))) (+ 1.0 x))
             (if (<= t_1 2e+260) t_3 (/ t_2 (+ 1.0 x))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double t_2 = (y / t) + x;
	double t_3 = (z * y) / (fma(z, t, -x) * (1.0 + x));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y / ((1.0 + x) * t);
	} else if (t_1 <= -4.0) {
		tmp = t_3;
	} else if (t_1 <= -5e-168) {
		tmp = ((y - (x / z)) / t) / 1.0;
	} else if (t_1 <= 5e-46) {
		tmp = t_2 / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / fma(t, z, -x))) / (1.0 + x);
	} else if (t_1 <= 2e+260) {
		tmp = t_3;
	} else {
		tmp = t_2 / (1.0 + x);
	}
	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(1.0 + x))
	t_2 = Float64(Float64(y / t) + x)
	t_3 = Float64(Float64(z * y) / Float64(fma(z, t, Float64(-x)) * Float64(1.0 + x)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y / Float64(Float64(1.0 + x) * t));
	elseif (t_1 <= -4.0)
		tmp = t_3;
	elseif (t_1 <= -5e-168)
		tmp = Float64(Float64(Float64(y - Float64(x / z)) / t) / 1.0);
	elseif (t_1 <= 5e-46)
		tmp = Float64(t_2 / 1.0);
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(1.0 + x));
	elseif (t_1 <= 2e+260)
		tmp = t_3;
	else
		tmp = Float64(t_2 / Float64(1.0 + x));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * y), $MachinePrecision] / N[(N[(z * t + (-x)), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4.0], t$95$3, If[LessEqual[t$95$1, -5e-168], N[(N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 5e-46], N[(t$95$2 / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+260], t$95$3, N[(t$95$2 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 37.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 75.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e260

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-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. +-commutative N/A

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

      12. lower-+.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 96.7%

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

   if -4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000001e-168

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity 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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.9%

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

   if -5.00000000000000001e-168 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999992e-46

   1. Initial program 89.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.0%

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

   if 4.99999999999999992e-46 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 2.00000000000000013e260 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 19.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 95.4

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

   5. Applied rewrites 95.4%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 6 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -\infty:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -4:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ t_2 := \frac{\frac{y}{t} + x}{1 + x}\\ t_3 := \frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;t\_1 \leq -4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 0.999999933177982:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x)))
        (t_2 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_3 (/ (* z y) (* (fma z t (- x)) (+ 1.0 x)))))
   (if (<= t_1 (- INFINITY))
     (/ y (* (+ 1.0 x) t))
     (if (<= t_1 -4.0)
       t_3
       (if (<= t_1 -5e-168)
         (/ (/ (- y (/ x z)) t) 1.0)
         (if (<= t_1 0.999999933177982)
           t_2
           (if (<= t_1 2.0) 1.0 (if (<= t_1 2e+260) t_3 t_2))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double t_2 = ((y / t) + x) / (1.0 + x);
	double t_3 = (z * y) / (fma(z, t, -x) * (1.0 + x));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y / ((1.0 + x) * t);
	} else if (t_1 <= -4.0) {
		tmp = t_3;
	} else if (t_1 <= -5e-168) {
		tmp = ((y - (x / z)) / t) / 1.0;
	} else if (t_1 <= 0.999999933177982) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= 2e+260) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	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(1.0 + x))
	t_2 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	t_3 = Float64(Float64(z * y) / Float64(fma(z, t, Float64(-x)) * Float64(1.0 + x)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y / Float64(Float64(1.0 + x) * t));
	elseif (t_1 <= -4.0)
		tmp = t_3;
	elseif (t_1 <= -5e-168)
		tmp = Float64(Float64(Float64(y - Float64(x / z)) / t) / 1.0);
	elseif (t_1 <= 0.999999933177982)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= 2e+260)
		tmp = t_3;
	else
		tmp = t_2;
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * y), $MachinePrecision] / N[(N[(z * t + (-x)), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4.0], t$95$3, If[LessEqual[t$95$1, -5e-168], N[(N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 0.999999933177982], t$95$2, If[LessEqual[t$95$1, 2.0], 1.0, If[LessEqual[t$95$1, 2e+260], t$95$3, t$95$2]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 37.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 75.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e260

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-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. +-commutative N/A

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

      12. lower-+.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 96.7%

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

   if -4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000001e-168

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity 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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.9%

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

   if -5.00000000000000001e-168 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999993317798197 or 2.00000000000000013e260 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 65.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if 0.99999993317798197 < (/.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 0

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

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -\infty:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -4:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 0.999999933177982:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{\frac{z}{t\_2} \cdot y}{1 + x}\\ t_4 := \frac{y}{t} + x\\ \mathbf{if}\;t\_1 \leq -4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{t\_4}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x)))
        (t_2 (fma t z (- x)))
        (t_3 (/ (* (/ z t_2) y) (+ 1.0 x)))
        (t_4 (+ (/ y t) x)))
   (if (<= t_1 -4.0)
     t_3
     (if (<= t_1 -5e-168)
       (/ (/ (- y (/ x z)) t) (+ 1.0 x))
       (if (<= t_1 5e-46)
         (/ t_4 1.0)
         (if (<= t_1 2.0)
           (/ (- x (/ x t_2)) (+ 1.0 x))
           (if (<= t_1 INFINITY) t_3 (/ t_4 (+ 1.0 x)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double t_2 = fma(t, z, -x);
	double t_3 = ((z / t_2) * y) / (1.0 + x);
	double t_4 = (y / t) + x;
	double tmp;
	if (t_1 <= -4.0) {
		tmp = t_3;
	} else if (t_1 <= -5e-168) {
		tmp = ((y - (x / z)) / t) / (1.0 + x);
	} else if (t_1 <= 5e-46) {
		tmp = t_4 / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_4 / (1.0 + x);
	}
	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(1.0 + x))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(Float64(z / t_2) * y) / Float64(1.0 + x))
	t_4 = Float64(Float64(y / t) + x)
	tmp = 0.0
	if (t_1 <= -4.0)
		tmp = t_3;
	elseif (t_1 <= -5e-168)
		tmp = Float64(Float64(Float64(y - Float64(x / z)) / t) / Float64(1.0 + x));
	elseif (t_1 <= 5e-46)
		tmp = Float64(t_4 / 1.0);
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(1.0 + x));
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(t_4 / Float64(1.0 + x));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -4.0], t$95$3, If[LessEqual[t$95$1, -5e-168], N[(N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-46], N[(t$95$4 / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, N[(t$95$4 / N[(1.0 + x), $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))) < -4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 78.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if -4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000001e-168

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity 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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 86.9%

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

   if -5.00000000000000001e-168 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999992e-46

   1. Initial program 89.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.0%

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

   if 4.99999999999999992e-46 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -4:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{\frac{z}{t\_2} \cdot y}{1 + x}\\ t_4 := \frac{y}{t} + x\\ \mathbf{if}\;t\_1 \leq -4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{t\_4}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x)))
        (t_2 (fma t z (- x)))
        (t_3 (/ (* (/ z t_2) y) (+ 1.0 x)))
        (t_4 (+ (/ y t) x)))
   (if (<= t_1 -4.0)
     t_3
     (if (<= t_1 -5e-168)
       (/ (/ (- y (/ x z)) t) 1.0)
       (if (<= t_1 5e-46)
         (/ t_4 1.0)
         (if (<= t_1 2.0)
           (/ (- x (/ x t_2)) (+ 1.0 x))
           (if (<= t_1 INFINITY) t_3 (/ t_4 (+ 1.0 x)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double t_2 = fma(t, z, -x);
	double t_3 = ((z / t_2) * y) / (1.0 + x);
	double t_4 = (y / t) + x;
	double tmp;
	if (t_1 <= -4.0) {
		tmp = t_3;
	} else if (t_1 <= -5e-168) {
		tmp = ((y - (x / z)) / t) / 1.0;
	} else if (t_1 <= 5e-46) {
		tmp = t_4 / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_4 / (1.0 + x);
	}
	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(1.0 + x))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(Float64(z / t_2) * y) / Float64(1.0 + x))
	t_4 = Float64(Float64(y / t) + x)
	tmp = 0.0
	if (t_1 <= -4.0)
		tmp = t_3;
	elseif (t_1 <= -5e-168)
		tmp = Float64(Float64(Float64(y - Float64(x / z)) / t) / 1.0);
	elseif (t_1 <= 5e-46)
		tmp = Float64(t_4 / 1.0);
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(1.0 + x));
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(t_4 / Float64(1.0 + x));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -4.0], t$95$3, If[LessEqual[t$95$1, -5e-168], N[(N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 5e-46], N[(t$95$4 / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, N[(t$95$4 / N[(1.0 + x), $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))) < -4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 78.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if -4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000001e-168

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity 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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.9%

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

   if -5.00000000000000001e-168 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999992e-46

   1. Initial program 89.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.0%

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

   if 4.99999999999999992e-46 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -4:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} + x\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ t_3 := \mathsf{fma}\left(t, z, -x\right)\\ \mathbf{if}\;t\_2 \leq -4:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{t\_3}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{t\_1}{1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_3}}{1 + x}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y t) x))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x)))
        (t_3 (fma t z (- x))))
   (if (<= t_2 -4.0)
     (* (/ z (+ 1.0 x)) (/ y t_3))
     (if (<= t_2 -5e-168)
       (/ (/ (- y (/ x z)) t) 1.0)
       (if (<= t_2 5e-46)
         (/ t_1 1.0)
         (if (<= t_2 2.0)
           (/ (- x (/ x t_3)) (+ 1.0 x))
           (if (<= t_2 2e+260)
             (/ (* z y) (* (fma z t (- x)) (+ 1.0 x)))
             (/ t_1 (+ 1.0 x)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / t) + x;
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double t_3 = fma(t, z, -x);
	double tmp;
	if (t_2 <= -4.0) {
		tmp = (z / (1.0 + x)) * (y / t_3);
	} else if (t_2 <= -5e-168) {
		tmp = ((y - (x / z)) / t) / 1.0;
	} else if (t_2 <= 5e-46) {
		tmp = t_1 / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_3)) / (1.0 + x);
	} else if (t_2 <= 2e+260) {
		tmp = (z * y) / (fma(z, t, -x) * (1.0 + x));
	} else {
		tmp = t_1 / (1.0 + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) + x)
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(1.0 + x))
	t_3 = fma(t, z, Float64(-x))
	tmp = 0.0
	if (t_2 <= -4.0)
		tmp = Float64(Float64(z / Float64(1.0 + x)) * Float64(y / t_3));
	elseif (t_2 <= -5e-168)
		tmp = Float64(Float64(Float64(y - Float64(x / z)) / t) / 1.0);
	elseif (t_2 <= 5e-46)
		tmp = Float64(t_1 / 1.0);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_3)) / Float64(1.0 + x));
	elseif (t_2 <= 2e+260)
		tmp = Float64(Float64(z * y) / Float64(fma(z, t, Float64(-x)) * Float64(1.0 + x)));
	else
		tmp = Float64(t_1 / Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * z + (-x)), $MachinePrecision]}, If[LessEqual[t$95$2, -4.0], N[(N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-168], N[(N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 5e-46], N[(t$95$1 / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+260], N[(N[(z * y), $MachinePrecision] / N[(N[(z * t + (-x)), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 73.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      12. lower-+.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if -4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000001e-168

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity 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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.9%

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

   if -5.00000000000000001e-168 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999992e-46

   1. Initial program 89.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.0%

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

   if 4.99999999999999992e-46 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e260

   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. +-commutative N/A

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

      12. lower-+.f64 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 99.0%

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

   if 2.00000000000000013e260 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 19.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 95.4

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

   5. Applied rewrites 95.4%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 6 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -4:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{\mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{1 + x}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ t_3 := \frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-13}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.999999933177982:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x)))
        (t_3 (/ (* z y) (* (fma z t (- x)) (+ 1.0 x)))))
   (if (<= t_2 (- INFINITY))
     (/ y (* (+ 1.0 x) t))
     (if (<= t_2 -4e-13)
       t_3
       (if (<= t_2 0.999999933177982)
         t_1
         (if (<= t_2 2.0) 1.0 (if (<= t_2 2e+260) t_3 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double t_3 = (z * y) / (fma(z, t, -x) * (1.0 + x));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y / ((1.0 + x) * t);
	} else if (t_2 <= -4e-13) {
		tmp = t_3;
	} else if (t_2 <= 0.999999933177982) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 2e+260) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(1.0 + x))
	t_3 = Float64(Float64(z * y) / Float64(fma(z, t, Float64(-x)) * Float64(1.0 + x)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y / Float64(Float64(1.0 + x) * t));
	elseif (t_2 <= -4e-13)
		tmp = t_3;
	elseif (t_2 <= 0.999999933177982)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 2e+260)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * y), $MachinePrecision] / N[(N[(z * t + (-x)), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-13], t$95$3, If[LessEqual[t$95$2, 0.999999933177982], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, 2e+260], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 37.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 75.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.0000000000000001e-13 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e260

   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. +-commutative N/A

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

      12. lower-+.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 94.6%

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

   if -4.0000000000000001e-13 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999993317798197 or 2.00000000000000013e260 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if 0.99999993317798197 < (/.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 0

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

   5. Applied rewrites 98.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -\infty:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 0.999999933177982:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, t, -x\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{\frac{z}{t\_2} \cdot y}{1 + x}\\ \mathbf{if}\;t\_1 \leq -4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-11}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x)))
        (t_2 (fma t z (- x)))
        (t_3 (/ (* (/ z t_2) y) (+ 1.0 x))))
   (if (<= t_1 -4.0)
     t_3
     (if (<= t_1 1e-11)
       (/ (- x (/ (- (/ x z) y) t)) (+ 1.0 x))
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ 1.0 x))
         (if (<= t_1 INFINITY) t_3 (/ (+ (/ y t) x) (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double t_2 = fma(t, z, -x);
	double t_3 = ((z / t_2) * y) / (1.0 + x);
	double tmp;
	if (t_1 <= -4.0) {
		tmp = t_3;
	} else if (t_1 <= 1e-11) {
		tmp = (x - (((x / z) - y) / t)) / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	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(1.0 + x))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(Float64(z / t_2) * y) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -4.0)
		tmp = t_3;
	elseif (t_1 <= 1e-11)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(1.0 + x));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(1.0 + x));
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4.0], t$95$3, If[LessEqual[t$95$1, 1e-11], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 78.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if -4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999939e-12

   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

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity 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 99.2

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

   5. Applied rewrites 99.2%

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

   if 9.99999999999999939e-12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -4:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 10^{-11}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, -x\right)\\ t_2 := \frac{\frac{z}{t\_1} \cdot y}{1 + x}\\ t_3 := x - z \cdot y\\ t_4 := \frac{x - \frac{t\_3}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_4 \leq -4:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 10^{-11}:\\ \;\;\;\;\frac{x - \frac{t\_3}{t \cdot z}}{1 + x}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{1 + x}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma t z (- x)))
        (t_2 (/ (* (/ z t_1) y) (+ 1.0 x)))
        (t_3 (- x (* z y)))
        (t_4 (/ (- x (/ t_3 (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_4 -4.0)
     t_2
     (if (<= t_4 1e-11)
       (/ (- x (/ t_3 (* t z))) (+ 1.0 x))
       (if (<= t_4 2.0)
         (/ (- x (/ x t_1)) (+ 1.0 x))
         (if (<= t_4 INFINITY) t_2 (/ (+ (/ y t) x) (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, z, -x);
	double t_2 = ((z / t_1) * y) / (1.0 + x);
	double t_3 = x - (z * y);
	double t_4 = (x - (t_3 / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_4 <= -4.0) {
		tmp = t_2;
	} else if (t_4 <= 1e-11) {
		tmp = (x - (t_3 / (t * z))) / (1.0 + x);
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (1.0 + x);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(t, z, Float64(-x))
	t_2 = Float64(Float64(Float64(z / t_1) * y) / Float64(1.0 + x))
	t_3 = Float64(x - Float64(z * y))
	t_4 = Float64(Float64(x - Float64(t_3 / Float64(Float64(t * z) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= -4.0)
		tmp = t_2;
	elseif (t_4 <= 1e-11)
		tmp = Float64(Float64(x - Float64(t_3 / Float64(t * z))) / Float64(1.0 + x));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(1.0 + x));
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z / t$95$1), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x - N[(t$95$3 / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4.0], t$95$2, If[LessEqual[t$95$4, 1e-11], N[(N[(x - N[(t$95$3 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 78.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if -4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999939e-12

   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

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

      1. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if 9.99999999999999939e-12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -4:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 10^{-11}:\\ \;\;\;\;\frac{x - \frac{x - z \cdot y}{t \cdot z}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-180}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_1 -2e-102)
     (/ y t)
     (if (<= t_1 5e-180)
       (* (- 1.0 x) x)
       (if (<= t_1 5e-46) (/ y t) (if (<= t_1 2.0) 1.0 (/ y t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e-102) {
		tmp = y / t;
	} else if (t_1 <= 5e-180) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 5e-46) {
		tmp = y / t;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0d0 + x)
    if (t_1 <= (-2d-102)) then
        tmp = y / t
    else if (t_1 <= 5d-180) then
        tmp = (1.0d0 - x) * x
    else if (t_1 <= 5d-46) then
        tmp = y / t
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e-102) {
		tmp = y / t;
	} else if (t_1 <= 5e-180) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 5e-46) {
		tmp = y / t;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x)
	tmp = 0
	if t_1 <= -2e-102:
		tmp = y / t
	elif t_1 <= 5e-180:
		tmp = (1.0 - x) * x
	elif t_1 <= 5e-46:
		tmp = y / t
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e-102)
		tmp = Float64(y / t);
	elseif (t_1 <= 5e-180)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_1 <= 5e-46)
		tmp = Float64(y / t);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -2e-102)
		tmp = y / t;
	elseif (t_1 <= 5e-180)
		tmp = (1.0 - x) * x;
	elseif (t_1 <= 5e-46)
		tmp = y / t;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-102], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 5e-180], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 5e-46], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999987e-102 or 5.0000000000000001e-180 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999992e-46 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 53.8

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

   5. Applied rewrites 53.8%

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

   if -1.99999999999999987e-102 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-180

   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 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 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.0%

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

   if 4.99999999999999992e-46 < (/.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 0

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

   5. Applied rewrites 94.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 5 \cdot 10^{-180}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{1 + x}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_2 \leq 0.999999933177982:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, z\right) \cdot \frac{y}{\mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_2 0.999999933177982)
     t_1
     (if (<= t_2 2.0)
       1.0
       (if (<= t_2 2e+260) (* (fma (- z) x z) (/ y (fma t z (- x)))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 0.999999933177982) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 2e+260) {
		tmp = fma(-z, x, z) * (y / fma(t, z, -x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= 0.999999933177982)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 2e+260)
		tmp = Float64(fma(Float64(-z), x, z) * Float64(y / fma(t, z, Float64(-x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.999999933177982], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, 2e+260], N[(N[((-z) * x + z), $MachinePrecision] * N[(y / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999993317798197 or 2.00000000000000013e260 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

   if 0.99999993317798197 < (/.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 0

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

   5. Applied rewrites 98.7%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e260

   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. +-commutative N/A

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

      12. lower-+.f64 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 82.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 0.999999933177982:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, z\right) \cdot \frac{y}{\mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(1 + x\right) \cdot t}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.999999933177982:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (+ 1.0 x) t)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_2 -2e-102)
     t_1
     (if (<= t_2 0.999999933177982)
       (/ x (+ 1.0 x))
       (if (<= t_2 2.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= -2e-102) {
		tmp = t_1;
	} else if (t_2 <= 0.999999933177982) {
		tmp = x / (1.0 + x);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y / ((1.0d0 + x) * t)
    t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0d0 + x)
    if (t_2 <= (-2d-102)) then
        tmp = t_1
    else if (t_2 <= 0.999999933177982d0) then
        tmp = x / (1.0d0 + x)
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= -2e-102) {
		tmp = t_1;
	} else if (t_2 <= 0.999999933177982) {
		tmp = x / (1.0 + x);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / ((1.0 + x) * t)
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x)
	tmp = 0
	if t_2 <= -2e-102:
		tmp = t_1
	elif t_2 <= 0.999999933177982:
		tmp = x / (1.0 + x)
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(1.0 + x) * t))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= -2e-102)
		tmp = t_1;
	elseif (t_2 <= 0.999999933177982)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / ((1.0 + x) * t);
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= -2e-102)
		tmp = t_1;
	elseif (t_2 <= 0.999999933177982)
		tmp = x / (1.0 + x);
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(1.0 + x), $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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-102], t$95$1, If[LessEqual[t$95$2, 0.999999933177982], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999987e-102 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 58.6%

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

   if -1.99999999999999987e-102 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999993317798197

   1. Initial program 91.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 49.8

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

   5. Applied rewrites 49.8%

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

   if 0.99999993317798197 < (/.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 0

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

   5. Applied rewrites 98.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 0.999999933177982:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.999999933177982:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_1 -2e-102)
     (/ y t)
     (if (<= t_1 0.999999933177982)
       (/ x (+ 1.0 x))
       (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e-102) {
		tmp = y / t;
	} else if (t_1 <= 0.999999933177982) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0d0 + x)
    if (t_1 <= (-2d-102)) then
        tmp = y / t
    else if (t_1 <= 0.999999933177982d0) then
        tmp = x / (1.0d0 + x)
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e-102) {
		tmp = y / t;
	} else if (t_1 <= 0.999999933177982) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x)
	tmp = 0
	if t_1 <= -2e-102:
		tmp = y / t
	elif t_1 <= 0.999999933177982:
		tmp = x / (1.0 + x)
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e-102)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.999999933177982)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -2e-102)
		tmp = y / t;
	elseif (t_1 <= 0.999999933177982)
		tmp = x / (1.0 + x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-102], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.999999933177982], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999987e-102 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 51.9

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

   5. Applied rewrites 51.9%

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

   if -1.99999999999999987e-102 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999993317798197

   1. Initial program 91.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 49.8

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

   5. Applied rewrites 49.8%

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

   if 0.99999993317798197 < (/.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 0

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

   5. Applied rewrites 98.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 0.999999933177982:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{\frac{1 + x}{z} \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x - \frac{\mathsf{fma}\left(z, y, -x\right)}{x - t \cdot z}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{\mathsf{fma}\left(t, x, t\right)} - \frac{x}{-1 - x}\right) - \frac{x}{\mathsf{fma}\left(z, x, z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_1 -2e+19)
     (/ y (* (/ (+ 1.0 x) z) (fma z t (- x))))
     (if (<= t_1 2e+260)
       (/ (- x (/ (fma z y (- x)) (- x (* t z)))) (+ 1.0 x))
       (- (- (/ y (fma t x t)) (/ x (- -1.0 x))) (/ x (* (fma z x z) t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = y / (((1.0 + x) / z) * fma(z, t, -x));
	} else if (t_1 <= 2e+260) {
		tmp = (x - (fma(z, y, -x) / (x - (t * z)))) / (1.0 + x);
	} else {
		tmp = ((y / fma(t, x, t)) - (x / (-1.0 - x))) - (x / (fma(z, x, z) * 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(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = Float64(y / Float64(Float64(Float64(1.0 + x) / z) * fma(z, t, Float64(-x))));
	elseif (t_1 <= 2e+260)
		tmp = Float64(Float64(x - Float64(fma(z, y, Float64(-x)) / Float64(x - Float64(t * z)))) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(Float64(y / fma(t, x, t)) - Float64(x / Float64(-1.0 - x))) - Float64(x / Float64(fma(z, x, z) * 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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+19], N[(y / N[(N[(N[(1.0 + x), $MachinePrecision] / z), $MachinePrecision] * N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+260], N[(N[(x - N[(N[(z * y + (-x)), $MachinePrecision] / N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision] - N[(x / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(z * x + z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e19

   1. Initial program 72.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 99.7%

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

   if -2e19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e260

   1. Initial program 98.2%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 98.2

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

   4. Applied rewrites 98.2%

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

   if 2.00000000000000013e260 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 19.7%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 95.5

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

   5. Applied rewrites 95.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{\frac{1 + x}{z} \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x - \frac{\mathsf{fma}\left(z, y, -x\right)}{x - t \cdot z}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{\mathsf{fma}\left(t, x, t\right)} - \frac{x}{-1 - x}\right) - \frac{x}{\mathsf{fma}\left(z, x, z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{\frac{1 + x}{z} \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x - \frac{\mathsf{fma}\left(z, y, -x\right)}{x - t \cdot z}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_1 -2e+19)
     (/ y (* (/ (+ 1.0 x) z) (fma z t (- x))))
     (if (<= t_1 2e+260)
       (/ (- x (/ (fma z y (- x)) (- x (* t z)))) (+ 1.0 x))
       (/ (+ (/ y t) x) (+ 1.0 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = y / (((1.0 + x) / z) * fma(z, t, -x));
	} else if (t_1 <= 2e+260) {
		tmp = (x - (fma(z, y, -x) / (x - (t * z)))) / (1.0 + x);
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	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(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = Float64(y / Float64(Float64(Float64(1.0 + x) / z) * fma(z, t, Float64(-x))));
	elseif (t_1 <= 2e+260)
		tmp = Float64(Float64(x - Float64(fma(z, y, Float64(-x)) / Float64(x - Float64(t * z)))) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+19], N[(y / N[(N[(N[(1.0 + x), $MachinePrecision] / z), $MachinePrecision] * N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+260], N[(N[(x - N[(N[(z * y + (-x)), $MachinePrecision] / N[(x - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $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))) < -2e19

   1. Initial program 72.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 99.7%

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

   if -2e19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e260

   1. Initial program 98.2%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 98.2

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

   4. Applied rewrites 98.2%

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

   if 2.00000000000000013e260 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 19.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{\frac{1 + x}{z} \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x - \frac{\mathsf{fma}\left(z, y, -x\right)}{x - t \cdot z}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{\frac{1 + x}{z} \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_1 -2e+19)
     (/ y (* (/ (+ 1.0 x) z) (fma z t (- x))))
     (if (<= t_1 2e+260) t_1 (/ (+ (/ y t) x) (+ 1.0 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = y / (((1.0 + x) / z) * fma(z, t, -x));
	} else if (t_1 <= 2e+260) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	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(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = Float64(y / Float64(Float64(Float64(1.0 + x) / z) * fma(z, t, Float64(-x))));
	elseif (t_1 <= 2e+260)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+19], N[(y / N[(N[(N[(1.0 + x), $MachinePrecision] / z), $MachinePrecision] * N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+260], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $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))) < -2e19

   1. Initial program 72.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 99.7%

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

   if -2e19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e260

   1. Initial program 98.2%

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

   if 2.00000000000000013e260 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 19.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{\frac{1 + x}{z} \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{1 + x}\\ t_2 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_2 \leq 0.999999933177982:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.00002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_2 0.999999933177982) t_1 (if (<= t_2 1.00002) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 0.999999933177982) {
		tmp = t_1;
	} else if (t_2 <= 1.00002) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((y / t) + x) / (1.0d0 + x)
    t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0d0 + x)
    if (t_2 <= 0.999999933177982d0) then
        tmp = t_1
    else if (t_2 <= 1.00002d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 0.999999933177982) {
		tmp = t_1;
	} else if (t_2 <= 1.00002) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (1.0 + x)
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x)
	tmp = 0
	if t_2 <= 0.999999933177982:
		tmp = t_1
	elif t_2 <= 1.00002:
		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(1.0 + x))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= 0.999999933177982)
		tmp = t_1;
	elseif (t_2 <= 1.00002)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (1.0 + x);
	t_2 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= 0.999999933177982)
		tmp = t_1;
	elseif (t_2 <= 1.00002)
		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[(1.0 + x), $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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.999999933177982], t$95$1, If[LessEqual[t$95$2, 1.00002], 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))) < 0.99999993317798197 or 1.00001999999999991 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 77.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if 0.99999993317798197 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00001999999999991

   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 0

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

   5. Applied rewrites 99.2%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 0.999999933177982:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 1.00002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 81.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x))))
   (if (<= t_1 5e-11)
     (/ (+ (/ y t) x) 1.0)
     (if (<= t_1 2.0) 1.0 (/ y (* (+ 1.0 x) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= 5e-11) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / ((1.0 + x) * 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))) / (1.0d0 + x)
    if (t_1 <= 5d-11) then
        tmp = ((y / t) + x) / 1.0d0
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / ((1.0d0 + x) * 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))) / (1.0 + x);
	double tmp;
	if (t_1 <= 5e-11) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / ((1.0 + x) * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x)
	tmp = 0
	if t_1 <= 5e-11:
		tmp = ((y / t) + x) / 1.0
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / ((1.0 + x) * 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(1.0 + x))
	tmp = 0.0
	if (t_1 <= 5e-11)
		tmp = Float64(Float64(Float64(y / t) + x) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / Float64(Float64(1.0 + x) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= 5e-11)
		tmp = ((y / t) + x) / 1.0;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / ((1.0 + x) * 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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-11], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / N[(N[(1.0 + x), $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))) < 5.00000000000000018e-11

   1. Initial program 86.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.3%

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

   if 5.00000000000000018e-11 < (/.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 0

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

   5. Applied rewrites 98.0%

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

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

   1. Initial program 60.4%

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

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

      12. lower-+.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 59.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (- x (/ (- x (* z y)) (- (* t z) x))) (+ 1.0 x)) 5e-11)
   (* (- 1.0 x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x)) <= 5e-11) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x - ((x - (z * y)) / ((t * z) - x))) / (1.0d0 + x)) <= 5d-11) then
        tmp = (1.0d0 - x) * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x)) <= 5e-11) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x)) <= 5e-11:
		tmp = (1.0 - x) * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(t * z) - x))) / Float64(1.0 + x)) <= 5e-11)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x - ((x - (z * y)) / ((t * z) - x))) / (1.0 + x)) <= 5e-11)
		tmp = (1.0 - x) * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 5e-11], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000018e-11

   1. Initial program 86.6%

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

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

      3. lower-+.f64 25.3

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.6%

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

   if 5.00000000000000018e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.7%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 75.6%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{t \cdot z - x}}{1 + x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 53.8% 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 88.7%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 52.0%

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

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024257 
(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)))