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

Percentage Accurate: 89.0% → 97.2%

Time: 12.4s

Alternatives: 15

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ t_2 := \mathsf{fma}\left(z, t, -x\right)\\ t_3 := \frac{z}{t\_2} \cdot \frac{y}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x + \frac{-x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)))
        (t_2 (fma z t (- x)))
        (t_3 (* (/ z t_2) (/ y (+ x 1.0)))))
   (if (<= t_1 -5e+21)
     t_3
     (if (<= t_1 5e-7)
       (/ (- x (/ (- (/ x z) y) t)) (+ x 1.0))
       (if (<= t_1 2.0)
         (/ (+ x (/ (- x) t_2)) (+ x 1.0))
         (if (<= t_1 INFINITY) t_3 (/ (+ x (/ y t)) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double t_2 = fma(z, t, -x);
	double t_3 = (z / t_2) * (y / (x + 1.0));
	double tmp;
	if (t_1 <= -5e+21) {
		tmp = t_3;
	} else if (t_1 <= 5e-7) {
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0);
	} else if (t_1 <= 2.0) {
		tmp = (x + (-x / t_2)) / (x + 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	t_2 = fma(z, t, Float64(-x))
	t_3 = Float64(Float64(z / t_2) * Float64(y / Float64(x + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e+21)
		tmp = t_3;
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x + 1.0));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x + Float64(Float64(-x) / t_2)) / Float64(x + 1.0));
	elseif (t_1 <= Inf)
		tmp = t_3;
	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[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * t + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / t$95$2), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+21], t$95$3, If[LessEqual[t$95$1, 5e-7], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x + N[((-x) / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, 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))) < -5e21 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 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

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 99.3%

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

   if -5e21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999977e-7

   1. Initial program 95.6%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 4.99999999999999977e-7 < (/.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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{-x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ t_2 := \mathsf{fma}\left(z, t, -x\right)\\ t_3 := \frac{z}{t\_2} \cdot \frac{y}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x + \frac{-x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)))
        (t_2 (fma z t (- x)))
        (t_3 (* (/ z t_2) (/ y (+ x 1.0)))))
   (if (<= t_1 -5e+21)
     t_3
     (if (<= t_1 5e-7)
       (/ (+ x (/ (- (* y z) x) (* z t))) (+ x 1.0))
       (if (<= t_1 2.0)
         (/ (+ x (/ (- x) t_2)) (+ x 1.0))
         (if (<= t_1 INFINITY) t_3 (/ (+ x (/ y t)) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double t_2 = fma(z, t, -x);
	double t_3 = (z / t_2) * (y / (x + 1.0));
	double tmp;
	if (t_1 <= -5e+21) {
		tmp = t_3;
	} else if (t_1 <= 5e-7) {
		tmp = (x + (((y * z) - x) / (z * t))) / (x + 1.0);
	} else if (t_1 <= 2.0) {
		tmp = (x + (-x / t_2)) / (x + 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	t_2 = fma(z, t, Float64(-x))
	t_3 = Float64(Float64(z / t_2) * Float64(y / Float64(x + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e+21)
		tmp = t_3;
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(z * t))) / Float64(x + 1.0));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x + Float64(Float64(-x) / t_2)) / Float64(x + 1.0));
	elseif (t_1 <= Inf)
		tmp = t_3;
	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[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * t + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / t$95$2), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+21], t$95$3, If[LessEqual[t$95$1, 5e-7], N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x + N[((-x) / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, 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))) < -5e21 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 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

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 99.3%

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

   if -5e21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999977e-7

   1. Initial program 95.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if 4.99999999999999977e-7 < (/.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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{-x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ t_3 := \mathsf{fma}\left(z, t, -x\right)\\ t_4 := \frac{z}{t\_3} \cdot \frac{y}{x + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x + \frac{-x}{t\_3}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)))
        (t_3 (fma z t (- x)))
        (t_4 (* (/ z t_3) (/ y (+ x 1.0)))))
   (if (<= t_2 -5e+21)
     t_4
     (if (<= t_2 5e-7)
       t_1
       (if (<= t_2 2.0)
         (/ (+ x (/ (- x) t_3)) (+ x 1.0))
         (if (<= t_2 INFINITY) 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 + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double t_3 = fma(z, t, -x);
	double t_4 = (z / t_3) * (y / (x + 1.0));
	double tmp;
	if (t_2 <= -5e+21) {
		tmp = t_4;
	} else if (t_2 <= 5e-7) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = (x + (-x / t_3)) / (x + 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		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(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	t_3 = fma(z, t, Float64(-x))
	t_4 = Float64(Float64(z / t_3) * Float64(y / Float64(x + 1.0)))
	tmp = 0.0
	if (t_2 <= -5e+21)
		tmp = t_4;
	elseif (t_2 <= 5e-7)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x + Float64(Float64(-x) / t_3)) / Float64(x + 1.0));
	elseif (t_2 <= Inf)
		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[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * t + (-x)), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z / t$95$3), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+21], t$95$4, If[LessEqual[t$95$2, 5e-7], 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, Infinity], 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))) < -5e21 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 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

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 99.3%

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

   if -5e21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999977e-7 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 4.99999999999999977e-7 < (/.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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{-x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ t_2 := \frac{x + \frac{y}{t}}{x + 1}\\ t_3 := x - z \cdot t\\ t_4 := \frac{x + \frac{x - y \cdot z}{t\_3}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{t\_3}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (fma z t (- x))) (/ y (+ x 1.0))))
        (t_2 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_3 (- x (* z t)))
        (t_4 (/ (+ x (/ (- x (* y z)) t_3)) (+ x 1.0))))
   (if (<= t_4 -5e+21)
     t_1
     (if (<= t_4 5e-7)
       t_2
       (if (<= t_4 2.0)
         (/ (+ x (/ x t_3)) (+ x 1.0))
         (if (<= t_4 INFINITY) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / fma(z, t, -x)) * (y / (x + 1.0));
	double t_2 = (x + (y / t)) / (x + 1.0);
	double t_3 = x - (z * t);
	double t_4 = (x + ((x - (y * z)) / t_3)) / (x + 1.0);
	double tmp;
	if (t_4 <= -5e+21) {
		tmp = t_1;
	} else if (t_4 <= 5e-7) {
		tmp = t_2;
	} else if (t_4 <= 2.0) {
		tmp = (x + (x / t_3)) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / fma(z, t, Float64(-x))) * Float64(y / Float64(x + 1.0)))
	t_2 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_3 = Float64(x - Float64(z * t))
	t_4 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / t_3)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -5e+21)
		tmp = t_1;
	elseif (t_4 <= 5e-7)
		tmp = t_2;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x + Float64(x / t_3)) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+21], t$95$1, If[LessEqual[t$95$4, 5e-7], t$95$2, If[LessEqual[t$95$4, 2.0], N[(N[(x + N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e21 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 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

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 99.3%

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

   if -5e21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999977e-7 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 4.99999999999999977e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{\left(x + 1\right) \cdot \mathsf{fma}\left(z, t, -x\right)}\\ t_2 := \frac{x + \frac{y}{t}}{x + 1}\\ t_3 := x - z \cdot t\\ t_4 := \frac{x + \frac{x - y \cdot z}{t\_3}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{t\_3}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ z (* (+ x 1.0) (fma z t (- x))))))
        (t_2 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_3 (- x (* z t)))
        (t_4 (/ (+ x (/ (- x (* y z)) t_3)) (+ x 1.0))))
   (if (<= t_4 -5e+21)
     t_1
     (if (<= t_4 5e-7)
       t_2
       (if (<= t_4 2.0)
         (/ (+ x (/ x t_3)) (+ x 1.0))
         (if (<= t_4 INFINITY) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (z / ((x + 1.0) * fma(z, t, -x)));
	double t_2 = (x + (y / t)) / (x + 1.0);
	double t_3 = x - (z * t);
	double t_4 = (x + ((x - (y * z)) / t_3)) / (x + 1.0);
	double tmp;
	if (t_4 <= -5e+21) {
		tmp = t_1;
	} else if (t_4 <= 5e-7) {
		tmp = t_2;
	} else if (t_4 <= 2.0) {
		tmp = (x + (x / t_3)) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(z / Float64(Float64(x + 1.0) * fma(z, t, Float64(-x)))))
	t_2 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_3 = Float64(x - Float64(z * t))
	t_4 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / t_3)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -5e+21)
		tmp = t_1;
	elseif (t_4 <= 5e-7)
		tmp = t_2;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x + Float64(x / t_3)) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / N[(N[(x + 1.0), $MachinePrecision] * N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+21], t$95$1, If[LessEqual[t$95$4, 5e-7], t$95$2, If[LessEqual[t$95$4, 2.0], N[(N[(x + N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e21 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 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

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 96.2%

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

   if -5e21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999977e-7 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 4.99999999999999977e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{z}{\left(x + 1\right) \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\left(x + 1\right) \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{\left(x + 1\right) \cdot \mathsf{fma}\left(z, t, -x\right)}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ t_3 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ z (* (+ x 1.0) (fma z t (- x))))))
        (t_2 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)))
        (t_3 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t_2 -5e+21)
     t_1
     (if (<= t_2 5e-7)
       t_3
       (if (<= t_2 2.0) 1.0 (if (<= t_2 INFINITY) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (z / ((x + 1.0) * fma(z, t, -x)));
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double t_3 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+21) {
		tmp = t_1;
	} else if (t_2 <= 5e-7) {
		tmp = t_3;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(z / Float64(Float64(x + 1.0) * fma(z, t, Float64(-x)))))
	t_2 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	t_3 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5e+21)
		tmp = t_1;
	elseif (t_2 <= 5e-7)
		tmp = t_3;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / N[(N[(x + 1.0), $MachinePrecision] * N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+21], t$95$1, If[LessEqual[t$95$2, 5e-7], t$95$3, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, Infinity], 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))) < -5e21 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 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

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 96.2%

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

   if -5e21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999977e-7 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 4.99999999999999977e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.7%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{z}{\left(x + 1\right) \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\left(x + 1\right) \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{\left(x + 1\right) \cdot \mathsf{fma}\left(z, t, -x\right)}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ t_3 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ y (* (+ x 1.0) (fma z t (- x))))))
        (t_2 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)))
        (t_3 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t_2 -1e-46)
     t_1
     (if (<= t_2 5e-7)
       t_3
       (if (<= t_2 2.0) 1.0 (if (<= t_2 INFINITY) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (y / ((x + 1.0) * fma(z, t, -x)));
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double t_3 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t_2 <= -1e-46) {
		tmp = t_1;
	} else if (t_2 <= 5e-7) {
		tmp = t_3;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(y / Float64(Float64(x + 1.0) * fma(z, t, Float64(-x)))))
	t_2 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	t_3 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1e-46)
		tmp = t_1;
	elseif (t_2 <= 5e-7)
		tmp = t_3;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-46], t$95$1, If[LessEqual[t$95$2, 5e-7], t$95$3, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, Infinity], 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))) < -1.00000000000000002e-46 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 85.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 85.5%

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

   if -1.00000000000000002e-46 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999977e-7 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if 4.99999999999999977e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{y}{\left(x + 1\right) \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\left(x + 1\right) \cdot \mathsf{fma}\left(z, t, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t \cdot \left(x + 1\right)}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* t (+ x 1.0))))
        (t_2 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_2 -5e-227)
     t_1
     (if (<= t_2 1e-30) (fma x (- x) x) (if (<= t_2 2.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (x + 1.0));
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e-227) {
		tmp = t_1;
	} else if (t_2 <= 1e-30) {
		tmp = fma(x, -x, x);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(t * Float64(x + 1.0)))
	t_2 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5e-227)
		tmp = t_1;
	elseif (t_2 <= 1e-30)
		tmp = fma(x, Float64(-x), x);
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-227], t$95$1, If[LessEqual[t$95$2, 1e-30], N[(x * (-x) + x), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.99999999999999961e-227 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 56.5%

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

   if -4.99999999999999961e-227 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-30

   1. Initial program 95.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 86.5%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 65.9

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.9%

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

   if 1e-30 < (/.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 95.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -5 \cdot 10^{-227}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-227}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_1 -5e-227)
     (/ y t)
     (if (<= t_1 1e-30) (fma x (- x) x) (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e-227) {
		tmp = y / t;
	} else if (t_1 <= 1e-30) {
		tmp = fma(x, -x, x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -5e-227)
		tmp = Float64(y / t);
	elseif (t_1 <= 1e-30)
		tmp = fma(x, Float64(-x), x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-227], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 1e-30], N[(x * (-x) + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.99999999999999961e-227 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 54.1

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

   5. Applied rewrites 54.1%

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

   if -4.99999999999999961e-227 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-30

   1. Initial program 95.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 86.5%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 65.9

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.9%

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

   if 1e-30 < (/.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 95.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -5 \cdot 10^{-227}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ t_2 := \mathsf{fma}\left(z, t, -x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+140}:\\ \;\;\;\;\frac{z}{t\_2} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, x, t\right)} + \left(\frac{x}{x + 1} - \frac{x}{t \cdot \mathsf{fma}\left(x, z, z\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)))
        (t_2 (fma z t (- x))))
   (if (<= t_1 -4e+140)
     (* (/ z t_2) (/ y (+ x 1.0)))
     (if (<= t_1 4e+262)
       (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))
       (+ (/ y (fma t x t)) (- (/ x (+ x 1.0)) (/ x (* t (fma x z z)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double t_2 = fma(z, t, -x);
	double tmp;
	if (t_1 <= -4e+140) {
		tmp = (z / t_2) * (y / (x + 1.0));
	} else if (t_1 <= 4e+262) {
		tmp = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	} else {
		tmp = (y / fma(t, x, t)) + ((x / (x + 1.0)) - (x / (t * fma(x, z, z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	t_2 = fma(z, t, Float64(-x))
	tmp = 0.0
	if (t_1 <= -4e+140)
		tmp = Float64(Float64(z / t_2) * Float64(y / Float64(x + 1.0)));
	elseif (t_1 <= 4e+262)
		tmp = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(y / fma(t, x, t)) + Float64(Float64(x / Float64(x + 1.0)) - 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[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * t + (-x)), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+140], N[(N[(z / t$95$2), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+262], N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision] + N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(t * N[(x * z + z), $MachinePrecision]), $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))) < -4.00000000000000024e140

   1. Initial program 74.6%

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

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 99.8%

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

   if -4.00000000000000024e140 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.0000000000000001e262

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if 4.0000000000000001e262 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 35.1%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-rgt-in N/A

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

      17. *-lft-identity N/A

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

      18. lower-fma.f64 86.0

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

   5. Applied rewrites 86.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -4 \cdot 10^{+140}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, x, t\right)} + \left(\frac{x}{x + 1} - \frac{x}{t \cdot \mathsf{fma}\left(x, z, z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ t_2 := \mathsf{fma}\left(z, t, -x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+140}:\\ \;\;\;\;\frac{z}{t\_2} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)))
        (t_2 (fma z t (- x))))
   (if (<= t_1 -4e+140)
     (* (/ z t_2) (/ y (+ x 1.0)))
     (if (<= t_1 4e+262)
       (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))
       (/ (+ x (/ y t)) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double t_2 = fma(z, t, -x);
	double tmp;
	if (t_1 <= -4e+140) {
		tmp = (z / t_2) * (y / (x + 1.0));
	} else if (t_1 <= 4e+262) {
		tmp = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	t_2 = fma(z, t, Float64(-x))
	tmp = 0.0
	if (t_1 <= -4e+140)
		tmp = Float64(Float64(z / t_2) * Float64(y / Float64(x + 1.0)));
	elseif (t_1 <= 4e+262)
		tmp = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * t + (-x)), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+140], N[(N[(z / t$95$2), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+262], N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000024e140

   1. Initial program 74.6%

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

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 99.8%

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

   if -4.00000000000000024e140 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.0000000000000001e262

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if 4.0000000000000001e262 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 35.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 85.8

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

   5. Applied rewrites 85.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -4 \cdot 10^{+140}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+140}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+262}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_1 -4e+140)
     (* (/ z (fma z t (- x))) (/ y (+ x 1.0)))
     (if (<= t_1 4e+262) t_1 (/ (+ x (/ y t)) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_1 <= -4e+140) {
		tmp = (z / fma(z, t, -x)) * (y / (x + 1.0));
	} else if (t_1 <= 4e+262) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -4e+140)
		tmp = Float64(Float64(z / fma(z, t, Float64(-x))) * Float64(y / Float64(x + 1.0)));
	elseif (t_1 <= 4e+262)
		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[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+140], N[(N[(z / N[(z * t + (-x)), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+262], 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))) < -4.00000000000000024e140

   1. Initial program 74.6%

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

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 99.8%

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

   if -4.00000000000000024e140 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.0000000000000001e262

   1. Initial program 99.0%

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

   if 4.0000000000000001e262 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 35.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 85.8

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

   5. Applied rewrites 85.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq -4 \cdot 10^{+140}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(z, t, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_2 5e-7) t_1 (if (<= t_2 1.05) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_2 <= 5e-7) {
		tmp = t_1;
	} else if (t_2 <= 1.05) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0d0)
    if (t_2 <= 5d-7) then
        tmp = t_1
    else if (t_2 <= 1.05d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_2 <= 5e-7) {
		tmp = t_1;
	} else if (t_2 <= 1.05) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0)
	tmp = 0
	if t_2 <= 5e-7:
		tmp = t_1
	elif t_2 <= 1.05:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 5e-7)
		tmp = t_1;
	elseif (t_2 <= 1.05)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 5e-7)
		tmp = t_1;
	elseif (t_2 <= 1.05)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-7], t$95$1, If[LessEqual[t$95$2, 1.05], 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))) < 4.99999999999999977e-7 or 1.05000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if 4.99999999999999977e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.05000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 1.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0)) 1e-30)
   (fma x (- x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0)) <= 1e-30) {
		tmp = fma(x, -x, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0)) <= 1e-30)
		tmp = fma(x, Float64(-x), x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1e-30], 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))) < 1e-30

   1. Initial program 90.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.0

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

   7. Applied rewrites 29.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 29.4%

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

   if 1e-30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 92.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 76.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.9% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 55.9%

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

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024223 
(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)))