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

Percentage Accurate: 89.4% → 97.9%

Time: 12.1s

Alternatives: 15

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_2 := \frac{x}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\frac{y}{x - z \cdot t}}{-1 - x}, t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \frac{y}{\mathsf{fma}\left(t, x, t\right)}\right) - \frac{x}{t \cdot \mathsf{fma}\left(x, z, z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_2 (/ x (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (fma z (/ (/ y (- x (* z t))) (- -1.0 x)) t_2)
     (if (<= t_1 5e+292)
       t_1
       (- (+ t_2 (/ y (fma t x t))) (/ x (* t (fma x z z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_2 = x / (x + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(z, ((y / (x - (z * t))) / (-1.0 - x)), t_2);
	} else if (t_1 <= 5e+292) {
		tmp = t_1;
	} else {
		tmp = (t_2 + (y / fma(t, x, t))) - (x / (t * fma(x, z, z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_2 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(z, Float64(Float64(y / Float64(x - Float64(z * t))) / Float64(-1.0 - x)), t_2);
	elseif (t_1 <= 5e+292)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_2 + Float64(y / fma(t, x, t))) - Float64(x / Float64(t * fma(x, z, z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(N[(y / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 5e+292], t$95$1, N[(N[(t$95$2 + N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(t * N[(x * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.9

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

   7. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-frac-neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

   9. Applied rewrites 99.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999996e292

   1. Initial program 98.5%

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

   if 4.9999999999999996e292 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 20.7%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. lower-fma.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\frac{y}{x - z \cdot t}}{-1 - x}, \frac{x}{x + 1}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(t, x, t\right)}\right) - \frac{x}{t \cdot \mathsf{fma}\left(x, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, -x\right)\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 0.0005:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma t z (- x)))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 -2e+14)
     (/ (+ x (/ (* y z) t_2)) (+ x 1.0))
     (if (<= t_3 0.0005)
       (/ (+ x (/ (- y (/ x z)) t)) (+ x 1.0))
       (if (<= t_3 2.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_3 5e+292)
           (/ (* y z) (* (+ x 1.0) t_1))
           (/ (+ x (/ y t)) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, z, -x);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -2e+14) {
		tmp = (x + ((y * z) / t_2)) / (x + 1.0);
	} else if (t_3 <= 0.0005) {
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= 5e+292) {
		tmp = (y * z) / ((x + 1.0) * t_1);
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(t, z, Float64(-x))
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -2e+14)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t_2)) / Float64(x + 1.0));
	elseif (t_3 <= 0.0005)
		tmp = Float64(Float64(x + Float64(Float64(y - Float64(x / z)) / t)) / Float64(x + 1.0));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_3 <= 5e+292)
		tmp = Float64(Float64(y * z) / Float64(Float64(x + 1.0) * t_1));
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+14], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0005], N[(N[(x + N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+292], N[(N[(y * z), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e14

   1. Initial program 84.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-4

   1. Initial program 94.1%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 5.0000000000000001e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999996e292

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 4.9999999999999996e292 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 20.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{z \cdot t - x}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0005:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_3 := \mathsf{fma}\left(t, z, -x\right)\\ t_4 := \frac{y \cdot z}{\left(x + 1\right) \cdot t\_3}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.0005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_3}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_3 (fma t z (- x)))
        (t_4 (/ (* y z) (* (+ x 1.0) t_3))))
   (if (<= t_2 -2e+14)
     t_4
     (if (<= t_2 0.0005)
       t_1
       (if (<= t_2 2.0)
         (/ (- x (/ x t_3)) (+ x 1.0))
         (if (<= t_2 5e+292) t_4 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_3 = fma(t, z, -x);
	double t_4 = (y * z) / ((x + 1.0) * t_3);
	double tmp;
	if (t_2 <= -2e+14) {
		tmp = t_4;
	} else if (t_2 <= 0.0005) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_3)) / (x + 1.0);
	} else if (t_2 <= 5e+292) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_3 = fma(t, z, Float64(-x))
	t_4 = Float64(Float64(y * z) / Float64(Float64(x + 1.0) * t_3))
	tmp = 0.0
	if (t_2 <= -2e+14)
		tmp = t_4;
	elseif (t_2 <= 0.0005)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_3)) / Float64(x + 1.0));
	elseif (t_2 <= 5e+292)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * z), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+14], t$95$4, If[LessEqual[t$95$2, 0.0005], t$95$1, If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+292], t$95$4, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999996e292

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-4 or 4.9999999999999996e292 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 5.0000000000000001e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0005:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_3 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.0005:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) (* (+ x 1.0) (fma t z (- x)))))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_3 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t_2 -2e+14)
     t_1
     (if (<= t_2 0.0005)
       t_3
       (if (<= t_2 2.0) 1.0 (if (<= t_2 5e+292) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / ((x + 1.0) * fma(t, z, -x));
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_3 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+14) {
		tmp = t_1;
	} else if (t_2 <= 0.0005) {
		tmp = t_3;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+292) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / Float64(Float64(x + 1.0) * fma(t, z, Float64(-x))))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_3 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e+14)
		tmp = t_1;
	elseif (t_2 <= 0.0005)
		tmp = t_3;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+292)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+14], t$95$1, If[LessEqual[t$95$2, 0.0005], t$95$3, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, 5e+292], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999996e292

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -2e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-4 or 4.9999999999999996e292 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 5.0000000000000001e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0005:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, -x\right)\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq 0.0005:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma t z (- x)))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 0.0005)
     (/ (+ x (/ (* y z) t_2)) (+ x 1.0))
     (if (<= t_3 2.0)
       (/ (- x (/ x t_1)) (+ x 1.0))
       (if (<= t_3 5e+292)
         (/ (* y z) (* (+ x 1.0) t_1))
         (/ (+ x (/ y t)) (+ x 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, z, -x);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= 0.0005) {
		tmp = (x + ((y * z) / t_2)) / (x + 1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= 5e+292) {
		tmp = (y * z) / ((x + 1.0) * t_1);
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(t, z, Float64(-x))
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= 0.0005)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t_2)) / Float64(x + 1.0));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_3 <= 5e+292)
		tmp = Float64(Float64(y * z) / Float64(Float64(x + 1.0) * t_1));
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0005], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+292], N[(N[(y * z), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-4

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if 5.0000000000000001e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999996e292

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 4.9999999999999996e292 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 20.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0005:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{z \cdot t - x}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -2e-43)
     t_1
     (if (<= t_2 2e-5) (- x (/ x (* z t))) (if (<= t_2 2e+27) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e-43) {
		tmp = t_1;
	} else if (t_2 <= 2e-5) {
		tmp = x - (x / (z * t));
	} else if (t_2 <= 2e+27) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e-43)
		tmp = t_1;
	elseif (t_2 <= 2e-5)
		tmp = Float64(x - Float64(x / Float64(z * t)));
	elseif (t_2 <= 2e+27)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-43], t$95$1, If[LessEqual[t$95$2, 2e-5], N[(x - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+27], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000015e-43 or 2e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.6%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 67.8

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

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. lower-fma.f64 67.1

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

   11. Applied rewrites 67.1%

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

   if -2.00000000000000015e-43 < (/.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 92.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-+.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. distribute-neg-frac N/A

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

       5. metadata-eval N/A

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

       6. associate-/l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r/ N/A

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

       9. mul-1-neg N/A

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

       10. unsub-neg N/A

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

       11. *-rgt-identity N/A

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

       12. associate-*r/ N/A

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

       13. lower--.f64 N/A

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

       14. associate-*r/ N/A

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

       15. *-rgt-identity N/A

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

       16. lower-/.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 69.8

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

   11. Applied rewrites 69.8%

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

   if 2.00000000000000016e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x - \frac{x}{z \cdot t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -2e-43)
     t_1
     (if (<= t_2 2e-5) (/ x (+ x 1.0)) (if (<= t_2 2e+27) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e-43) {
		tmp = t_1;
	} else if (t_2 <= 2e-5) {
		tmp = x / (x + 1.0);
	} else if (t_2 <= 2e+27) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e-43)
		tmp = t_1;
	elseif (t_2 <= 2e-5)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_2 <= 2e+27)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-43], t$95$1, If[LessEqual[t$95$2, 2e-5], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+27], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000015e-43 or 2e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.6%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 67.8

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

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. lower-fma.f64 67.1

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

   11. Applied rewrites 67.1%

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

   if -2.00000000000000015e-43 < (/.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 92.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   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 66.4

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

   5. Applied rewrites 66.4%

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

   if 2.00000000000000016e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.9%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -2e-43)
     (/ y t)
     (if (<= t_1 2e-5) (/ x (+ x 1.0)) (if (<= t_1 2e+27) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e-43) {
		tmp = y / t;
	} else if (t_1 <= 2e-5) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 2e+27) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= (-2d-43)) then
        tmp = y / t
    else if (t_1 <= 2d-5) then
        tmp = x / (x + 1.0d0)
    else if (t_1 <= 2d+27) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e-43) {
		tmp = y / t;
	} else if (t_1 <= 2e-5) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 2e+27) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -2e-43:
		tmp = y / t
	elif t_1 <= 2e-5:
		tmp = x / (x + 1.0)
	elif t_1 <= 2e+27:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -2e-43)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-5)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_1 <= 2e+27)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -2e-43)
		tmp = y / t;
	elseif (t_1 <= 2e-5)
		tmp = x / (x + 1.0);
	elseif (t_1 <= 2e+27)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-43], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-5], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+27], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000015e-43 or 2e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if -2.00000000000000015e-43 < (/.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 92.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   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 66.4

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

   5. Applied rewrites 66.4%

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

   if 2.00000000000000016e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -2e-43)
     (/ y t)
     (if (<= t_1 2e-5) (fma x (- x) x) (if (<= t_1 2e+27) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e-43) {
		tmp = y / t;
	} else if (t_1 <= 2e-5) {
		tmp = fma(x, -x, x);
	} else if (t_1 <= 2e+27) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -2e-43)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-5)
		tmp = fma(x, Float64(-x), x);
	elseif (t_1 <= 2e+27)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-43], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-5], N[(x * (-x) + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+27], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000015e-43 or 2e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if -2.00000000000000015e-43 < (/.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 92.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   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 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 65.6

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

   8. Applied rewrites 65.6%

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

   if 2.00000000000000016e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\frac{y}{x - z \cdot t}}{-1 - x}, \frac{x}{x + 1}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (fma z (/ (/ y (- x (* z t))) (- -1.0 x)) (/ x (+ x 1.0)))
     (if (<= t_1 5e+292) t_1 (/ (+ x (/ y t)) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(z, ((y / (x - (z * t))) / (-1.0 - x)), (x / (x + 1.0)));
	} else if (t_1 <= 5e+292) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(z, Float64(Float64(y / Float64(x - Float64(z * t))) / Float64(-1.0 - x)), Float64(x / Float64(x + 1.0)));
	elseif (t_1 <= 5e+292)
		tmp = t_1;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(N[(y / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+292], t$95$1, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.9

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

   7. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-frac-neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

   9. Applied rewrites 99.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999996e292

   1. Initial program 98.5%

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

   if 4.9999999999999996e292 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 20.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\frac{y}{x - z \cdot t}}{-1 - x}, \frac{x}{x + 1}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 0.0005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 0.0005) t_1 (if (<= t_2 2e+27) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.0005) {
		tmp = t_1;
	} else if (t_2 <= 2e+27) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_2 <= 0.0005d0) then
        tmp = t_1
    else if (t_2 <= 2d+27) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.0005) {
		tmp = t_1;
	} else if (t_2 <= 2e+27) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= 0.0005:
		tmp = t_1
	elif t_2 <= 2e+27:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 0.0005)
		tmp = t_1;
	elseif (t_2 <= 2e+27)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 0.0005)
		tmp = t_1;
	elseif (t_2 <= 2e+27)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0005], t$95$1, If[LessEqual[t$95$2, 2e+27], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-4 or 2e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 5.0000000000000001e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.6%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0005:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 5e+292) t_1 (/ (+ x (/ y t)) (+ x 1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e+292) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= 5d+292) then
        tmp = t_1
    else
        tmp = (x + (y / t)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e+292) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= 5e+292:
		tmp = t_1
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= 5e+292)
		tmp = t_1;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= 5e+292)
		tmp = t_1;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+292], t$95$1, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999996e292

   1. Initial program 96.1%

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

   if 4.9999999999999996e292 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 20.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   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 35.2

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

   5. Applied rewrites 35.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 32.1

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

   8. Applied rewrites 32.1%

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

   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. Initial program 85.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 77.7%

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

4. Final simplification 62.1%

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

Alternative 14: 63.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 2e-5], x, 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))) < 2.00000000000000016e-5

   1. Initial program 90.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 38.2

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-+.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 34.4

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

   8. Applied rewrites 34.4%

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

   9. Taylor expanded in t around inf

      \[\leadsto x \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 31.5%

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

   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. Initial program 85.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 77.7%

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

4. Final simplification 61.9%

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

Alternative 15: 54.2% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 1.0)

C

double code(double x, double y, double z, double t) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return 1.0

Julia

function code(x, y, z, t)
	return 1.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 87.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 52.6%

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

Developer Target 1: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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)))