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

Percentage Accurate: 89.3% → 96.8%

Time: 9.8s

Alternatives: 16

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{\mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+254}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 (- INFINITY))
     (* (/ z (+ 1.0 x)) (/ y (fma t z (- x))))
     (if (<= t_1 5e+254) t_1 (/ (+ (/ y t) x) (+ 1.0 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (z / (1.0 + x)) * (y / fma(t, z, -x));
	} else if (t_1 <= 5e+254) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(z / Float64(1.0 + x)) * Float64(y / fma(t, z, Float64(-x))));
	elseif (t_1 <= 5e+254)
		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[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+254], 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))) < -inf.0

   1. Initial program 16.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999994e254

   1. Initial program 99.4%

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

   if 4.99999999999999994e254 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 36.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -\infty:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{\mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ t_2 := \frac{\frac{y}{t}}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+28}:\\ \;\;\;\;1 - \frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot y\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999822:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x)))
        (t_2 (/ (/ y t) (+ 1.0 x))))
   (if (<= t_1 -2e+28)
     (- 1.0 (* (/ z (fma x x x)) y))
     (if (<= t_1 -5e-6)
       t_2
       (if (<= t_1 0.9999999999999822)
         (/ x (+ 1.0 x))
         (if (<= t_1 2.0) 1.0 (if (<= t_1 INFINITY) t_2 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double t_2 = (y / t) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e+28) {
		tmp = 1.0 - ((z / fma(x, x, x)) * y);
	} else if (t_1 <= -5e-6) {
		tmp = t_2;
	} else if (t_1 <= 0.9999999999999822) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(y / t) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e+28)
		tmp = Float64(1.0 - Float64(Float64(z / fma(x, x, x)) * y));
	elseif (t_1 <= -5e-6)
		tmp = t_2;
	elseif (t_1 <= 0.9999999999999822)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / t), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+28], N[(1.0 - N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-6], t$95$2, If[LessEqual[t$95$1, 0.9999999999999822], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, If[LessEqual[t$95$1, Infinity], t$95$2, 1.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999992e28

   1. Initial program 66.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x + {x}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.5%

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

   if -1.99999999999999992e28 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000041e-6 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 87.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if -5.00000000000000041e-6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.9999999999999822

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 0.9999999999999822 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{+28}:\\ \;\;\;\;1 - \frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{y}{t}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.9999999999999822:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{y}{t}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+28}:\\ \;\;\;\;1 - \frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot y\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999822:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -2e+28)
     (- 1.0 (* (/ z (fma x x x)) y))
     (if (<= t_1 -5e-6)
       (/ y t)
       (if (<= t_1 0.9999999999999822)
         (/ x (+ 1.0 x))
         (if (<= t_1 2.0) 1.0 (if (<= t_1 INFINITY) (/ y t) 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e+28) {
		tmp = 1.0 - ((z / fma(x, x, x)) * y);
	} else if (t_1 <= -5e-6) {
		tmp = y / t;
	} else if (t_1 <= 0.9999999999999822) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e+28)
		tmp = Float64(1.0 - Float64(Float64(z / fma(x, x, x)) * y));
	elseif (t_1 <= -5e-6)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.9999999999999822)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+28], N[(1.0 - N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-6], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999822], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, If[LessEqual[t$95$1, Infinity], N[(y / t), $MachinePrecision], 1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999992e28

   1. Initial program 66.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x + {x}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.5%

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

   if -1.99999999999999992e28 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000041e-6 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 87.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -5.00000000000000041e-6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.9999999999999822

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 0.9999999999999822 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{+28}:\\ \;\;\;\;1 - \frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.9999999999999822:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+28}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999822:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -2e+28)
     (* (- y) (/ z (fma x x x)))
     (if (<= t_1 -5e-6)
       (/ y t)
       (if (<= t_1 0.9999999999999822)
         (/ x (+ 1.0 x))
         (if (<= t_1 2.0) 1.0 (if (<= t_1 INFINITY) (/ y t) 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e+28) {
		tmp = -y * (z / fma(x, x, x));
	} else if (t_1 <= -5e-6) {
		tmp = y / t;
	} else if (t_1 <= 0.9999999999999822) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e+28)
		tmp = Float64(Float64(-y) * Float64(z / fma(x, x, x)));
	elseif (t_1 <= -5e-6)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.9999999999999822)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+28], N[((-y) * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-6], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999822], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, If[LessEqual[t$95$1, Infinity], N[(y / t), $MachinePrecision], 1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999992e28

   1. Initial program 66.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot z}{x + {x}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.8%

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

   10. Applied rewrites 48.2%

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

   if -1.99999999999999992e28 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000041e-6 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 87.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -5.00000000000000041e-6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.9999999999999822

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 0.9999999999999822 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{+28}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.9999999999999822:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, -x\right)\\ t_2 := \frac{\frac{z}{t\_1} \cdot y}{1 + x}\\ t_3 := z \cdot y - x\\ t_4 := \frac{\frac{t\_3}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_4 \leq -5000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t\_3}{t \cdot z} + x}{1 + x}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{1 + x}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+254}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

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 (- (* z y) x))
        (t_4 (/ (+ (/ t_3 (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_4 -5000000.0)
     t_2
     (if (<= t_4 2e-5)
       (/ (+ (/ t_3 (* t z)) x) (+ 1.0 x))
       (if (<= t_4 2.0)
         (/ (- x (/ x t_1)) (+ 1.0 x))
         (if (<= t_4 5e+254) 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 = (z * y) - x;
	double t_4 = ((t_3 / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_4 <= -5000000.0) {
		tmp = t_2;
	} else if (t_4 <= 2e-5) {
		tmp = ((t_3 / (t * z)) + x) / (1.0 + x);
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (1.0 + x);
	} else if (t_4 <= 5e+254) {
		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(Float64(z * y) - x)
	t_4 = Float64(Float64(Float64(t_3 / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= -5000000.0)
		tmp = t_2;
	elseif (t_4 <= 2e-5)
		tmp = Float64(Float64(Float64(t_3 / Float64(t * z)) + x) / Float64(1.0 + x));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(1.0 + x));
	elseif (t_4 <= 5e+254)
		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[(N[(z * y), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$3 / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5000000.0], t$95$2, If[LessEqual[t$95$4, 2e-5], N[(N[(N[(t$95$3 / N[(t * z), $MachinePrecision]), $MachinePrecision] + x), $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, 5e+254], 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))) < -5e6 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999994e254

   1. Initial program 80.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -5e6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000016e-5

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 2.00000000000000016e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   if 4.99999999999999994e254 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 36.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5000000:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{t \cdot z} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{\frac{z}{t\_2} \cdot y}{1 + x}\\ t_4 := \frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+254}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x)))
        (t_2 (fma t z (- x)))
        (t_3 (/ (* (/ z t_2) y) (+ 1.0 x)))
        (t_4 (/ (+ (/ y t) x) (+ 1.0 x))))
   (if (<= t_1 -5000000.0)
     t_3
     (if (<= t_1 2e-5)
       t_4
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ 1.0 x))
         (if (<= t_1 5e+254) t_3 t_4))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + 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) / (1.0 + x);
	double tmp;
	if (t_1 <= -5000000.0) {
		tmp = t_3;
	} else if (t_1 <= 2e-5) {
		tmp = t_4;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_1 <= 5e+254) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + 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(Float64(y / t) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5000000.0)
		tmp = t_3;
	elseif (t_1 <= 2e-5)
		tmp = t_4;
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(1.0 + x));
	elseif (t_1 <= 5e+254)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $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[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000.0], t$95$3, If[LessEqual[t$95$1, 2e-5], t$95$4, 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, 5e+254], t$95$3, t$95$4]]]]]]]]

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))) < -5e6 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999994e254

   1. Initial program 80.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -5e6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000016e-5 or 4.99999999999999994e254 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   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 z around inf

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

      1. lower-/.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if 2.00000000000000016e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5000000:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{z}{1 + x} \cdot \frac{y}{t\_2}\\ t_4 := \frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x)))
        (t_2 (fma t z (- x)))
        (t_3 (* (/ z (+ 1.0 x)) (/ y t_2)))
        (t_4 (/ (+ (/ y t) x) (+ 1.0 x))))
   (if (<= t_1 -5000000.0)
     t_3
     (if (<= t_1 2e-5)
       t_4
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ 1.0 x))
         (if (<= t_1 INFINITY) t_3 t_4))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double t_2 = fma(t, z, -x);
	double t_3 = (z / (1.0 + x)) * (y / t_2);
	double t_4 = ((y / t) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5000000.0) {
		tmp = t_3;
	} else if (t_1 <= 2e-5) {
		tmp = t_4;
	} 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;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(z / Float64(1.0 + x)) * Float64(y / t_2))
	t_4 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5000000.0)
		tmp = t_3;
	elseif (t_1 <= 2e-5)
		tmp = t_4;
	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 = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000.0], t$95$3, If[LessEqual[t$95$1, 2e-5], t$95$4, 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, t$95$4]]]]]]]]

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))) < -5e6 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 77.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -5e6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000016e-5 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if 2.00000000000000016e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5000000:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{\mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{\mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+28}:\\ \;\;\;\;1 - \frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot y\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{y}{t}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -2e+28)
     (- 1.0 (* (/ z (fma x x x)) y))
     (if (<= t_1 0.0001)
       (/ (+ (/ y t) x) 1.0)
       (if (<= t_1 2.0)
         1.0
         (if (<= t_1 INFINITY) (/ (/ y t) (+ 1.0 x)) 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e+28) {
		tmp = 1.0 - ((z / fma(x, x, x)) * y);
	} else if (t_1 <= 0.0001) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / t) / (1.0 + x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e+28)
		tmp = Float64(1.0 - Float64(Float64(z / fma(x, x, x)) * y));
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(Float64(y / t) + x) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / t) / Float64(1.0 + x));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+28], N[(1.0 - N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, If[LessEqual[t$95$1, Infinity], N[(N[(y / t), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999992e28

   1. Initial program 66.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x + {x}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.5%

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

   if -1.99999999999999992e28 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 84.3

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.6%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 90.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.0%

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

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

   1. Initial program 84.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 67.9

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

   5. Applied rewrites 67.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{+28}:\\ \;\;\;\;1 - \frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.0001:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{y}{t}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999822:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -5e-6)
     (/ y t)
     (if (<= t_1 0.9999999999999822)
       (/ x (+ 1.0 x))
       (if (<= t_1 2.0) 1.0 (if (<= t_1 INFINITY) (/ y t) 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e-6) {
		tmp = y / t;
	} else if (t_1 <= 0.9999999999999822) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e-6) {
		tmp = y / t;
	} else if (t_1 <= 0.9999999999999822) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_1 <= -5e-6:
		tmp = y / t
	elif t_1 <= 0.9999999999999822:
		tmp = x / (1.0 + x)
	elif t_1 <= 2.0:
		tmp = 1.0
	elif t_1 <= math.inf:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e-6)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.9999999999999822)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -5e-6)
		tmp = y / t;
	elseif (t_1 <= 0.9999999999999822)
		tmp = x / (1.0 + x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-6], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999822], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, If[LessEqual[t$95$1, Infinity], N[(y / t), $MachinePrecision], 1.0]]]]]

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))) < -5.00000000000000041e-6 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 78.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   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 -5.00000000000000041e-6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.9999999999999822

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 0.9999999999999822 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.9999999999999822:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -5e-6)
     (/ y t)
     (if (<= t_1 0.0001)
       (* (fma (- x 1.0) x 1.0) x)
       (if (<= t_1 2.0) 1.0 (if (<= t_1 INFINITY) (/ y t) 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e-6) {
		tmp = y / t;
	} else if (t_1 <= 0.0001) {
		tmp = fma((x - 1.0), x, 1.0) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e-6)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.0001)
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-6], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, If[LessEqual[t$95$1, Infinity], N[(y / t), $MachinePrecision], 1.0]]]]]

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))) < -5.00000000000000041e-6 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 78.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   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 -5.00000000000000041e-6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.0%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 90.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -5e-6)
     (/ y t)
     (if (<= t_1 0.0001)
       (* (- 1.0 x) x)
       (if (<= t_1 2.0) 1.0 (if (<= t_1 INFINITY) (/ y t) 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e-6) {
		tmp = y / t;
	} else if (t_1 <= 0.0001) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e-6) {
		tmp = y / t;
	} else if (t_1 <= 0.0001) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_1 <= -5e-6:
		tmp = y / t
	elif t_1 <= 0.0001:
		tmp = (1.0 - x) * x
	elif t_1 <= 2.0:
		tmp = 1.0
	elif t_1 <= math.inf:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e-6)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -5e-6)
		tmp = y / t;
	elseif (t_1 <= 0.0001)
		tmp = (1.0 - x) * x;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-6], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, If[LessEqual[t$95$1, Infinity], N[(y / t), $MachinePrecision], 1.0]]]]]

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))) < -5.00000000000000041e-6 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 78.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   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 -5.00000000000000041e-6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 56.0

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

   5. Applied rewrites 56.0%

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

   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 56.0%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 90.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.0001:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ t_2 := \frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t - y}{1 + x}}{x}, z, 1\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.00000000002:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x)))
        (t_2 (/ (+ (/ y t) x) (+ 1.0 x))))
   (if (<= t_1 -5e+72)
     (fma (/ (/ (- t y) (+ 1.0 x)) x) z 1.0)
     (if (<= t_1 2e-5)
       t_2
       (if (<= t_1 1.00000000002)
         (/ (- x (/ x (fma t z (- x)))) (+ 1.0 x))
         t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double t_2 = ((y / t) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e+72) {
		tmp = fma((((t - y) / (1.0 + x)) / x), z, 1.0);
	} else if (t_1 <= 2e-5) {
		tmp = t_2;
	} else if (t_1 <= 1.00000000002) {
		tmp = (x - (x / fma(t, z, -x))) / (1.0 + x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e+72)
		tmp = fma(Float64(Float64(Float64(t - y) / Float64(1.0 + x)) / x), z, 1.0);
	elseif (t_1 <= 2e-5)
		tmp = t_2;
	elseif (t_1 <= 1.00000000002)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(1.0 + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $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]}, If[LessEqual[t$95$1, -5e+72], N[(N[(N[(N[(t - y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * z + 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2e-5], t$95$2, If[LessEqual[t$95$1, 1.00000000002], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.99999999999999992e72

   1. Initial program 57.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 49.9%

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

   7. Applied rewrites 61.3%

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

   if -4.99999999999999992e72 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000016e-5 or 1.00000000002 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 83.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 81.6

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

   5. Applied rewrites 81.6%

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

   if 2.00000000000000016e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000002

   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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t - y}{1 + x}}{x}, z, 1\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 1.00000000002:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ t_2 := \frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t - y}{1 + x}}{x}, z, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999822:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.00000000002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x)))
        (t_2 (/ (+ (/ y t) x) (+ 1.0 x))))
   (if (<= t_1 -5e+72)
     (fma (/ (/ (- t y) (+ 1.0 x)) x) z 1.0)
     (if (<= t_1 0.9999999999999822)
       t_2
       (if (<= t_1 1.00000000002) 1.0 t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double t_2 = ((y / t) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e+72) {
		tmp = fma((((t - y) / (1.0 + x)) / x), z, 1.0);
	} else if (t_1 <= 0.9999999999999822) {
		tmp = t_2;
	} else if (t_1 <= 1.00000000002) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e+72)
		tmp = fma(Float64(Float64(Float64(t - y) / Float64(1.0 + x)) / x), z, 1.0);
	elseif (t_1 <= 0.9999999999999822)
		tmp = t_2;
	elseif (t_1 <= 1.00000000002)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $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]}, If[LessEqual[t$95$1, -5e+72], N[(N[(N[(N[(t - y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * z + 1.0), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999822], t$95$2, If[LessEqual[t$95$1, 1.00000000002], 1.0, t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.99999999999999992e72

   1. Initial program 57.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 49.9%

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

   7. Applied rewrites 61.3%

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

   if -4.99999999999999992e72 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.9999999999999822 or 1.00000000002 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 84.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

   if 0.9999999999999822 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000002

   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 99.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t - y}{1 + x}}{x}, z, 1\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.9999999999999822:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 1.00000000002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ t_2 := \frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+28}:\\ \;\;\;\;1 - \frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot y\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999822:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.00000000002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x)))
        (t_2 (/ (+ (/ y t) x) (+ 1.0 x))))
   (if (<= t_1 -2e+28)
     (- 1.0 (* (/ z (fma x x x)) y))
     (if (<= t_1 0.9999999999999822)
       t_2
       (if (<= t_1 1.00000000002) 1.0 t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double t_2 = ((y / t) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e+28) {
		tmp = 1.0 - ((z / fma(x, x, x)) * y);
	} else if (t_1 <= 0.9999999999999822) {
		tmp = t_2;
	} else if (t_1 <= 1.00000000002) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e+28)
		tmp = Float64(1.0 - Float64(Float64(z / fma(x, x, x)) * y));
	elseif (t_1 <= 0.9999999999999822)
		tmp = t_2;
	elseif (t_1 <= 1.00000000002)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $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]}, If[LessEqual[t$95$1, -2e+28], N[(1.0 - N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999822], t$95$2, If[LessEqual[t$95$1, 1.00000000002], 1.0, t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999992e28

   1. Initial program 66.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x + {x}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.5%

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

   if -1.99999999999999992e28 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.9999999999999822 or 1.00000000002 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 83.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 82.8

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

   5. Applied rewrites 82.8%

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

   if 0.9999999999999822 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000002

   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 99.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{+28}:\\ \;\;\;\;1 - \frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.9999999999999822:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 1.00000000002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.0001:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x)) 0.0001)
   (* (- 1.0 x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 0.0001) {
		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 ((((((z * y) - x) / ((t * z) - x)) + x) / (1.0d0 + x)) <= 0.0001d0) 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 ((((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 0.0001) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 0.0001:
		tmp = (1.0 - x) * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x)) <= 0.0001)
		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 ((((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 0.0001)
		tmp = (1.0 - x) * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 0.0001], 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))) < 1.00000000000000005e-4

   1. Initial program 89.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 38.3

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

   5. Applied rewrites 38.3%

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

   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 36.9%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 79.5%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 0.0001:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.6% 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 89.3%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 53.8%

   \[\leadsto \color{blue}{1} \]
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 2024244 
(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)))