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

Percentage Accurate: 89.8% → 95.6%

Time: 11.7s

Alternatives: 17

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: 89.8% 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: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(t, x, t\right)}\right) - \frac{x}{t \cdot \mathsf{fma}\left(x, z, z\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+190}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (+ (/ x (+ x 1.0)) (/ y (fma t x t))) (/ x (* t (fma x z z)))))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 1e+190) t_2 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x / (x + 1.0)) + (y / fma(t, x, t))) - (x / (t * fma(x, z, z)));
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+190) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x / Float64(x + 1.0)) + Float64(y / fma(t, x, t))) - Float64(x / Float64(t * fma(x, z, z))))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+190)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(t * N[(x * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+190], t$95$2, 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))) < -inf.0 or 1.0000000000000001e190 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 22.7%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. lower-fma.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.0000000000000001e190

   1. Initial program 98.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\left(\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(t, x, t\right)}\right) - \frac{x}{t \cdot \mathsf{fma}\left(x, z, z\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+190}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(t, x, t\right)}\right) - \frac{x}{t \cdot \mathsf{fma}\left(x, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{y \cdot z}{\left(x + 1\right) \cdot t\_2}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq -20000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{+190}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_2 (fma t z (- x)))
        (t_3 (/ (* y z) (* (+ x 1.0) t_2))))
   (if (<= t_1 (- INFINITY))
     (/ (/ y t) (+ x 1.0))
     (if (<= t_1 -20000.0)
       t_3
       (if (<= t_1 0.5)
         (/ (+ x (/ (- y (/ x z)) t)) (+ x 1.0))
         (if (<= t_1 2.0)
           (/ (- x (/ x t_2)) (+ x 1.0))
           (if (<= t_1 1e+190) t_3 (/ (+ x (/ y t)) (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_2 = fma(t, z, -x);
	double t_3 = (y * z) / ((x + 1.0) * t_2);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / t) / (x + 1.0);
	} else if (t_1 <= -20000.0) {
		tmp = t_3;
	} else if (t_1 <= 0.5) {
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0);
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_1 <= 1e+190) {
		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(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(y * z) / Float64(Float64(x + 1.0) * t_2))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	elseif (t_1 <= -20000.0)
		tmp = t_3;
	elseif (t_1 <= 0.5)
		tmp = Float64(Float64(x + Float64(Float64(y - Float64(x / z)) / t)) / Float64(x + 1.0));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_1 <= 1e+190)
		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[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000.0], t$95$3, If[LessEqual[t$95$1, 0.5], N[(N[(x + N[(N[(y - N[(x / z), $MachinePrecision]), $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, 1e+190], t$95$3, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 24.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.0000000000000001e190

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -2e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 96.1%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 1.0000000000000001e190 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 21.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -20000:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.5:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+190}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{y \cdot z}{\left(x + 1\right) \cdot t\_2}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq -20000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{+190}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_2 (fma t z (- x)))
        (t_3 (/ (* y z) (* (+ x 1.0) t_2))))
   (if (<= t_1 (- INFINITY))
     (/ (/ y t) (+ x 1.0))
     (if (<= t_1 -20000.0)
       t_3
       (if (<= t_1 0.5)
         (/ (+ x (/ (- y (/ x z)) t)) 1.0)
         (if (<= t_1 2.0)
           (/ (- x (/ x t_2)) (+ x 1.0))
           (if (<= t_1 1e+190) t_3 (/ (+ x (/ y t)) (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_2 = fma(t, z, -x);
	double t_3 = (y * z) / ((x + 1.0) * t_2);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / t) / (x + 1.0);
	} else if (t_1 <= -20000.0) {
		tmp = t_3;
	} else if (t_1 <= 0.5) {
		tmp = (x + ((y - (x / z)) / t)) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_1 <= 1e+190) {
		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(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(y * z) / Float64(Float64(x + 1.0) * t_2))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	elseif (t_1 <= -20000.0)
		tmp = t_3;
	elseif (t_1 <= 0.5)
		tmp = Float64(Float64(x + Float64(Float64(y - Float64(x / z)) / t)) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_1 <= 1e+190)
		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[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -20000.0], t$95$3, If[LessEqual[t$95$1, 0.5], N[(N[(x + N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / 1.0), $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, 1e+190], t$95$3, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 24.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.0000000000000001e190

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -2e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 96.1%

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

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 96.1

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-lft-in N/A

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 1.0000000000000001e190 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 21.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -20000:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.5:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+190}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, -x\right)\\ t_2 := y \cdot z - x\\ t_3 := \frac{x + \frac{t\_2}{z \cdot t - x}}{x + 1}\\ t_4 := \frac{y \cdot z}{\left(x + 1\right) \cdot t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq -20000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.5:\\ \;\;\;\;\frac{x + \frac{t\_2}{z \cdot t}}{1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 10^{+190}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma t z (- x)))
        (t_2 (- (* y z) x))
        (t_3 (/ (+ x (/ t_2 (- (* z t) x))) (+ x 1.0)))
        (t_4 (/ (* y z) (* (+ x 1.0) t_1))))
   (if (<= t_3 (- INFINITY))
     (/ (/ y t) (+ x 1.0))
     (if (<= t_3 -20000.0)
       t_4
       (if (<= t_3 0.5)
         (/ (+ x (/ t_2 (* z t))) 1.0)
         (if (<= t_3 2.0)
           (/ (- x (/ x t_1)) (+ x 1.0))
           (if (<= t_3 1e+190) t_4 (/ (+ x (/ y t)) (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, z, -x);
	double t_2 = (y * z) - x;
	double t_3 = (x + (t_2 / ((z * t) - x))) / (x + 1.0);
	double t_4 = (y * z) / ((x + 1.0) * t_1);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (y / t) / (x + 1.0);
	} else if (t_3 <= -20000.0) {
		tmp = t_4;
	} else if (t_3 <= 0.5) {
		tmp = (x + (t_2 / (z * t))) / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= 1e+190) {
		tmp = t_4;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(t, z, Float64(-x))
	t_2 = Float64(Float64(y * z) - x)
	t_3 = Float64(Float64(x + Float64(t_2 / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_4 = Float64(Float64(y * z) / Float64(Float64(x + 1.0) * t_1))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	elseif (t_3 <= -20000.0)
		tmp = t_4;
	elseif (t_3 <= 0.5)
		tmp = Float64(Float64(x + Float64(t_2 / Float64(z * t))) / 1.0);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_3 <= 1e+190)
		tmp = t_4;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(t$95$2 / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * z), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -20000.0], t$95$4, If[LessEqual[t$95$3, 0.5], N[(N[(x + N[(t$95$2 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+190], t$95$4, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 24.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.0000000000000001e190

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -2e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 96.1%

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

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 96.1

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in t around -inf

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-lft-in N/A

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.5%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 92.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 1.0000000000000001e190 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 21.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -20000:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.5:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+190}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_3 := \mathsf{fma}\left(t, z, -x\right)\\ t_4 := \frac{y \cdot z}{\left(x + 1\right) \cdot t\_3}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq -20000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_3}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{+190}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_3 (fma t z (- x)))
        (t_4 (/ (* y z) (* (+ x 1.0) t_3))))
   (if (<= t_2 (- INFINITY))
     (/ (/ y t) (+ x 1.0))
     (if (<= t_2 -20000.0)
       t_4
       (if (<= t_2 0.5)
         t_1
         (if (<= t_2 2.0)
           (/ (- x (/ x t_3)) (+ x 1.0))
           (if (<= t_2 1e+190) t_4 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_3 = fma(t, z, -x);
	double t_4 = (y * z) / ((x + 1.0) * t_3);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / t) / (x + 1.0);
	} else if (t_2 <= -20000.0) {
		tmp = t_4;
	} else if (t_2 <= 0.5) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_3)) / (x + 1.0);
	} else if (t_2 <= 1e+190) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_3 = fma(t, z, Float64(-x))
	t_4 = Float64(Float64(y * z) / Float64(Float64(x + 1.0) * t_3))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	elseif (t_2 <= -20000.0)
		tmp = t_4;
	elseif (t_2 <= 0.5)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_3)) / Float64(x + 1.0));
	elseif (t_2 <= 1e+190)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * z), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -20000.0], t$95$4, If[LessEqual[t$95$2, 0.5], 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, 1e+190], t$95$4, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 24.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.0000000000000001e190

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -2e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.5 or 1.0000000000000001e190 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 88.1

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

   5. Applied rewrites 88.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}}{x + 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -20000:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.5:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+190}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_3 := \frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq -20000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 10^{+190}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_3 (/ (* y z) (* (+ x 1.0) (fma t z (- x))))))
   (if (<= t_2 (- INFINITY))
     (/ (/ y t) (+ x 1.0))
     (if (<= t_2 -20000.0)
       t_3
       (if (<= t_2 0.5)
         t_1
         (if (<= t_2 2.0) 1.0 (if (<= t_2 1e+190) t_3 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_3 = (y * z) / ((x + 1.0) * fma(t, z, -x));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / t) / (x + 1.0);
	} else if (t_2 <= -20000.0) {
		tmp = t_3;
	} else if (t_2 <= 0.5) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 1e+190) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_3 = Float64(Float64(y * z) / Float64(Float64(x + 1.0) * fma(t, z, Float64(-x))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	elseif (t_2 <= -20000.0)
		tmp = t_3;
	elseif (t_2 <= 0.5)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 1e+190)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -20000.0], t$95$3, If[LessEqual[t$95$2, 0.5], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, 1e+190], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 24.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e4 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.0000000000000001e190

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -2e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.5 or 1.0000000000000001e190 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 88.1

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

   5. Applied rewrites 88.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.1%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -20000:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.5:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+190}:\\ \;\;\;\;\frac{y \cdot z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_3 := \frac{z}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+113}:\\ \;\;\;\;t\_3 \cdot \left(-y\right)\\ \mathbf{elif}\;t\_2 \leq 0.5:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;t\_2 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+185}:\\ \;\;\;\;1 - y \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ y t) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_3 (/ z (fma x x x))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -2e+113)
       (* t_3 (- y))
       (if (<= t_2 0.5)
         (/ (+ x (/ y t)) 1.0)
         (if (<= t_2 1.2) 1.0 (if (<= t_2 5e+185) (- 1.0 (* y t_3)) 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 + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_3 = z / fma(x, x, x);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -2e+113) {
		tmp = t_3 * -y;
	} else if (t_2 <= 0.5) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_2 <= 1.2) {
		tmp = 1.0;
	} else if (t_2 <= 5e+185) {
		tmp = 1.0 - (y * t_3);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_3 = Float64(z / fma(x, x, x))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -2e+113)
		tmp = Float64(t_3 * Float64(-y));
	elseif (t_2 <= 0.5)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	elseif (t_2 <= 1.2)
		tmp = 1.0;
	elseif (t_2 <= 5e+185)
		tmp = Float64(1.0 - Float64(y * t_3));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -2e+113], N[(t$95$3 * (-y)), $MachinePrecision], If[LessEqual[t$95$2, 0.5], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 1.2], 1.0, If[LessEqual[t$95$2, 5e+185], N[(1.0 - N[(y * t$95$3), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 4.9999999999999999e185 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 27.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e113

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.4%

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

   if -2e113 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 96.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

         \[\leadsto \frac{x + \frac{1}{t \cdot z - x} \cdot \color{blue}{\frac{{\left(y \cdot z\right)}^{3} - {x}^{3}}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) + \left(x \cdot x + \left(y \cdot z\right) \cdot x\right)}}}{x + 1} \]

      6. clear-num N/A

         \[\leadsto \frac{x + \frac{1}{t \cdot z - x} \cdot \color{blue}{\frac{1}{\frac{\left(y \cdot z\right) \cdot \left(y \cdot z\right) + \left(x \cdot x + \left(y \cdot z\right) \cdot x\right)}{{\left(y \cdot z\right)}^{3} - {x}^{3}}}}}{x + 1} \]

      7. clear-num N/A

         \[\leadsto \frac{x + \frac{1}{t \cdot z - x} \cdot \frac{1}{\color{blue}{\frac{1}{\frac{{\left(y \cdot z\right)}^{3} - {x}^{3}}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) + \left(x \cdot x + \left(y \cdot z\right) \cdot x\right)}}}}}{x + 1} \]

      8. flip3-- N/A

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

      9. lift--.f64 N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 96.6

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 84.4

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 80.6%

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

   if 0.5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   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 98.8%

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

   if 1.19999999999999996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e185

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 47.3

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.3%

      \[\leadsto 1 - \color{blue}{y \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+113}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot \left(-y\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.5:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;1 - y \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, 1 - x, -1\right), x\right)\\ \mathbf{elif}\;t\_2 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+185}:\\ \;\;\;\;1 - y \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ y t) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -2e+18)
     t_1
     (if (<= t_2 2e-7)
       (fma x (* x (fma x (- 1.0 x) -1.0)) x)
       (if (<= t_2 1.2)
         1.0
         (if (<= t_2 5e+185) (- 1.0 (* y (/ z (fma x x x)))) 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 + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+18) {
		tmp = t_1;
	} else if (t_2 <= 2e-7) {
		tmp = fma(x, (x * fma(x, (1.0 - x), -1.0)), x);
	} else if (t_2 <= 1.2) {
		tmp = 1.0;
	} else if (t_2 <= 5e+185) {
		tmp = 1.0 - (y * (z / fma(x, x, x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e+18)
		tmp = t_1;
	elseif (t_2 <= 2e-7)
		tmp = fma(x, Float64(x * fma(x, Float64(1.0 - x), -1.0)), x);
	elseif (t_2 <= 1.2)
		tmp = 1.0;
	elseif (t_2 <= 5e+185)
		tmp = Float64(1.0 - Float64(y * Float64(z / fma(x, x, x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+18], t$95$1, If[LessEqual[t$95$2, 2e-7], N[(x * N[(x * N[(x * N[(1.0 - x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 1.2], 1.0, If[LessEqual[t$95$2, 5e+185], N[(1.0 - N[(y * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e18 or 4.9999999999999999e185 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 46.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7

   1. Initial program 96.1%

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

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 96.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

   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. +-commutative N/A

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

      3. lower-+.f64 59.6

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

   7. Applied rewrites 59.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 59.6%

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

   if 1.9999999999999999e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   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 97.4%

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

   if 1.19999999999999996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e185

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 47.3

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.3%

      \[\leadsto 1 - \color{blue}{y \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, 1 - x, -1\right), x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;1 - y \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot \left(-y\right)\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -20000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{+15}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+185}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (fma x x x)) (- y)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -20000.0)
     t_1
     (if (<= t_2 2e-7)
       (/ x (+ x 1.0))
       (if (<= t_2 1e+15) 1.0 (if (<= t_2 5e+185) t_1 (/ y t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / fma(x, x, x)) * -y;
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -20000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e-7) {
		tmp = x / (x + 1.0);
	} else if (t_2 <= 1e+15) {
		tmp = 1.0;
	} else if (t_2 <= 5e+185) {
		tmp = t_1;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / fma(x, x, x)) * Float64(-y))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -20000.0)
		tmp = t_1;
	elseif (t_2 <= 2e-7)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_2 <= 1e+15)
		tmp = 1.0;
	elseif (t_2 <= 5e+185)
		tmp = t_1;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -20000.0], t$95$1, If[LessEqual[t$95$2, 2e-7], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+15], 1.0, If[LessEqual[t$95$2, 5e+185], t$95$1, N[(y / t), $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))) < -2e4 or 1e15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e185

   1. Initial program 76.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 48.4%

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

   if -2e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7

   1. Initial program 96.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if 1.9999999999999999e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e15

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.1%

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

   if 4.9999999999999999e185 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 28.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 51.5

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

   5. Applied rewrites 51.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -20000:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot \left(-y\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+15}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, 1 - x, -1\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -2e+18)
     (/ y t)
     (if (<= t_1 2e-7)
       (fma x (* x (fma x (- 1.0 x) -1.0)) x)
       (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e+18) {
		tmp = y / t;
	} else if (t_1 <= 2e-7) {
		tmp = fma(x, (x * fma(x, (1.0 - x), -1.0)), 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(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -2e+18)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-7)
		tmp = fma(x, Float64(x * fma(x, Float64(1.0 - x), -1.0)), 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[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+18], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[(x * N[(x * N[(x * N[(1.0 - x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 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))) < -2e18 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 55.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 43.8

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

   5. Applied rewrites 43.8%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7

   1. Initial program 96.1%

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

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 96.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

   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. +-commutative N/A

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

      3. lower-+.f64 59.6

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

   7. Applied rewrites 59.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 59.6%

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

   if 1.9999999999999999e-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 96.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, 1 - x, -1\right), x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -2e+18)
     (/ y t)
     (if (<= t_1 2e-7) (/ x (+ x 1.0)) (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e+18) {
		tmp = y / t;
	} else if (t_1 <= 2e-7) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= (-2d+18)) then
        tmp = y / t
    else if (t_1 <= 2d-7) then
        tmp = x / (x + 1.0d0)
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e+18) {
		tmp = y / t;
	} else if (t_1 <= 2e-7) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -2e+18:
		tmp = y / t
	elif t_1 <= 2e-7:
		tmp = x / (x + 1.0)
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -2e+18)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-7)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -2e+18)
		tmp = y / t;
	elseif (t_1 <= 2e-7)
		tmp = x / (x + 1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+18], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e18 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 55.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 43.8

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

   5. Applied rewrites 43.8%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 59.6

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

   5. Applied rewrites 59.6%

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

   if 1.9999999999999999e-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 96.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -2e+18)
     (/ y t)
     (if (<= t_1 2e-7) (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 + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e+18) {
		tmp = y / t;
	} else if (t_1 <= 2e-7) {
		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(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -2e+18)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-7)
		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[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+18], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], 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))) < -2e18 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 55.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 43.8

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

   5. Applied rewrites 43.8%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7

   1. Initial program 96.1%

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

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 96.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

   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. +-commutative N/A

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

      3. lower-+.f64 59.6

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

   7. Applied rewrites 59.6%

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

   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 58.8%

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

   if 1.9999999999999999e-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 96.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{+190}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (/ (/ y t) (+ x 1.0))
     (if (<= t_1 1e+190) t_1 (/ (+ x (/ y t)) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / t) / (x + 1.0);
	} else if (t_1 <= 1e+190) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / t) / (x + 1.0);
	} else if (t_1 <= 1e+190) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (y / t) / (x + 1.0)
	elif t_1 <= 1e+190:
		tmp = t_1
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	elseif (t_1 <= 1e+190)
		tmp = t_1;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (y / t) / (x + 1.0);
	elseif (t_1 <= 1e+190)
		tmp = t_1;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+190], t$95$1, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.0000000000000001e190

   1. Initial program 98.9%

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

   if 1.0000000000000001e190 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 21.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+190}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)) 2e-7)
   (fma x (- x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 2e-7) {
		tmp = fma(x, -x, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0)) <= 2e-7)
		tmp = fma(x, Float64(-x), x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 2e-7], N[(x * (-x) + x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.9999999999999999e-7

   1. Initial program 85.0%

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

      1. lift-+.f64 N/A

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

      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. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 85.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.0

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

   4. Applied rewrites 85.0%

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

   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. +-commutative N/A

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

      3. lower-+.f64 40.8

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

   7. Applied rewrites 40.8%

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

   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 38.6%

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

   if 1.9999999999999999e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 75.9%

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

4. Final simplification 63.5%

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

Alternative 15: 83.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-53}:\\ \;\;\;\;1 - y \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t -3.2e-61)
     t_1
     (if (<= t 2.4e-53) (- 1.0 (* y (/ z (fma x x x)))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t <= -3.2e-61) {
		tmp = t_1;
	} else if (t <= 2.4e-53) {
		tmp = 1.0 - (y * (z / fma(x, x, x)));
	} 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))
	tmp = 0.0
	if (t <= -3.2e-61)
		tmp = t_1;
	elseif (t <= 2.4e-53)
		tmp = Float64(1.0 - Float64(y * Float64(z / fma(x, x, x))));
	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]}, If[LessEqual[t, -3.2e-61], t$95$1, If[LessEqual[t, 2.4e-53], N[(1.0 - N[(y * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2000000000000001e-61 or 2.40000000000000007e-53 < t

   1. Initial program 84.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -3.2000000000000001e-61 < t < 2.40000000000000007e-53

   1. Initial program 88.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 81.8%

      \[\leadsto 1 - \color{blue}{y \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 70.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;t \leq -0.00138:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 10^{+44}:\\ \;\;\;\;1 - y \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ x 1.0))))
   (if (<= t -0.00138)
     t_1
     (if (<= t 1e+44) (- 1.0 (* y (/ z (fma x x x)))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double tmp;
	if (t <= -0.00138) {
		tmp = t_1;
	} else if (t <= 1e+44) {
		tmp = 1.0 - (y * (z / fma(x, x, x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (t <= -0.00138)
		tmp = t_1;
	elseif (t <= 1e+44)
		tmp = Float64(1.0 - Float64(y * Float64(z / fma(x, x, x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.00138], t$95$1, If[LessEqual[t, 1e+44], N[(1.0 - N[(y * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.00137999999999999993 or 1.0000000000000001e44 < t

   1. Initial program 83.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -0.00137999999999999993 < t < 1.0000000000000001e44

   1. Initial program 87.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 74.1%

      \[\leadsto 1 - \color{blue}{y \cdot \frac{z}{\mathsf{fma}\left(x, x, x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 53.3% 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 85.8%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 52.5%

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

Developer Target 1: 99.4% 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 2024238 
(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)))