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

Percentage Accurate: 88.6% → 97.6%

Time: 13.7s

Alternatives: 16

Speedup: 0.6×

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: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \mathsf{fma}\left(y, \frac{z}{t\_1 \cdot \left(x + 1\right)}, 1\right)\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -100000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.0001:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (fma y (/ z (* t_1 (+ x 1.0))) 1.0))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_3 -100000.0)
     t_2
     (if (<= t_3 0.0001)
       (/ (+ x (/ (- y (/ x z)) t)) (+ x 1.0))
       (if (<= t_3 INFINITY) t_2 (/ (+ x (/ y t)) (+ x 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = fma(y, (z / (t_1 * (x + 1.0))), 1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -100000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.0001) {
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = fma(y, Float64(z / Float64(t_1 * Float64(x + 1.0))), 1.0)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -100000.0)
		tmp = t_2;
	elseif (t_3 <= 0.0001)
		tmp = Float64(Float64(x + Float64(Float64(y - Float64(x / z)) / t)) / Float64(x + 1.0));
	elseif (t_3 <= Inf)
		tmp = t_2;
	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[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z / N[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -100000.0], t$95$2, If[LessEqual[t$95$3, 0.0001], N[(N[(x + N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, 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))) < -1e5 or 1.00000000000000005e-4 < (/.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 91.7%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 85.6

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

   5. Simplified 85.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 92.8

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

   8. Simplified 92.8%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 98.4%

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

   if -1e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 93.9%

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

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

      12. /-lowering-/.f64 99.1

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

   5. Simplified 99.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -100000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}, 1\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+195}:\\ \;\;\;\;\frac{x - y \cdot \frac{z}{x}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 0.0001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 10^{+285}:\\ \;\;\;\;\frac{y \cdot z}{t\_2 \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 -1e+195)
     (/ (- x (* y (/ z x))) (+ x 1.0))
     (if (<= t_3 0.0001)
       t_1
       (if (<= t_3 2.0)
         1.0
         (if (<= t_3 1e+285) (/ (* y z) (* t_2 (+ x 1.0))) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1e+195) {
		tmp = (x - (y * (z / x))) / (x + 1.0);
	} else if (t_3 <= 0.0001) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= 1e+285) {
		tmp = (y * z) / (t_2 * (x + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (z * t) - x
    t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0d0)
    if (t_3 <= (-1d+195)) then
        tmp = (x - (y * (z / x))) / (x + 1.0d0)
    else if (t_3 <= 0.0001d0) then
        tmp = t_1
    else if (t_3 <= 2.0d0) then
        tmp = 1.0d0
    else if (t_3 <= 1d+285) then
        tmp = (y * z) / (t_2 * (x + 1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1e+195) {
		tmp = (x - (y * (z / x))) / (x + 1.0);
	} else if (t_3 <= 0.0001) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= 1e+285) {
		tmp = (y * z) / (t_2 * (x + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (z * t) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -1e+195:
		tmp = (x - (y * (z / x))) / (x + 1.0)
	elif t_3 <= 0.0001:
		tmp = t_1
	elif t_3 <= 2.0:
		tmp = 1.0
	elif t_3 <= 1e+285:
		tmp = (y * z) / (t_2 * (x + 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -1e+195)
		tmp = Float64(Float64(x - Float64(y * Float64(z / x))) / Float64(x + 1.0));
	elseif (t_3 <= 0.0001)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= 1e+285)
		tmp = Float64(Float64(y * z) / Float64(t_2 * Float64(x + 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (z * t) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -1e+195)
		tmp = (x - (y * (z / x))) / (x + 1.0);
	elseif (t_3 <= 0.0001)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= 1e+285)
		tmp = (y * z) / (t_2 * (x + 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+195], N[(N[(x - N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0001], t$95$1, If[LessEqual[t$95$3, 2.0], 1.0, If[LessEqual[t$95$3, 1e+285], N[(N[(y * z), $MachinePrecision] / N[(t$95$2 * N[(x + 1.0), $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))) < -9.99999999999999977e194

   1. Initial program 48.4%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 48.4

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

   5. Simplified 48.4%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 83.3

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

   8. Simplified 83.3%

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

   if -9.99999999999999977e194 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4 or 9.9999999999999998e284 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.7%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 82.3

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

   5. Simplified 82.3%

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

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

   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. Simplified 98.4%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999998e284

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 97.8

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

   5. Simplified 97.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+195}:\\ \;\;\;\;\frac{x - y \cdot \frac{z}{x}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+285}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;t\_3 \leq 0.0001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 10^{+285}:\\ \;\;\;\;\frac{y \cdot z}{t\_2 \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 -1e+195)
     (fma (- 0.0 y) (/ z (fma x x x)) 1.0)
     (if (<= t_3 0.0001)
       t_1
       (if (<= t_3 2.0)
         1.0
         (if (<= t_3 1e+285) (/ (* y z) (* t_2 (+ x 1.0))) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1e+195) {
		tmp = fma((0.0 - y), (z / fma(x, x, x)), 1.0);
	} else if (t_3 <= 0.0001) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= 1e+285) {
		tmp = (y * z) / (t_2 * (x + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -1e+195)
		tmp = fma(Float64(0.0 - y), Float64(z / fma(x, x, x)), 1.0);
	elseif (t_3 <= 0.0001)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= 1e+285)
		tmp = Float64(Float64(y * z) / Float64(t_2 * Float64(x + 1.0)));
	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[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+195], N[(N[(0.0 - y), $MachinePrecision] * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$3, 0.0001], t$95$1, If[LessEqual[t$95$3, 2.0], 1.0, If[LessEqual[t$95$3, 1e+285], N[(N[(y * z), $MachinePrecision] / N[(t$95$2 * N[(x + 1.0), $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))) < -9.99999999999999977e194

   1. Initial program 48.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out-- N/A

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

      9. distribute-lft-out-- N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 67.2%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. div-sub N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. div-sub N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 72.7

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

   8. Simplified 72.7%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. associate-/l* N/A

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

       4. distribute-lft-neg-in N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{0 - y}, \frac{z}{x \cdot \left(1 + x\right)}, 1\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{0 - y}, \frac{z}{x \cdot \left(1 + x\right)}, 1\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(0 - y, \color{blue}{\frac{z}{x \cdot \left(1 + x\right)}}, 1\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{x \cdot \color{blue}{\left(x + 1\right)}}, 1\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{\color{blue}{x \cdot x + x \cdot 1}}, 1\right) \]

       11. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{x \cdot x + \color{blue}{x}}, 1\right) \]

       12. accelerator-lowering-fma.f64 83.0

           \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}}, 1\right) \]

   11. Simplified 83.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)} \]

   if -9.99999999999999977e194 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4 or 9.9999999999999998e284 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.7%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 82.3

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

   5. Simplified 82.3%

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

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

   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. Simplified 98.4%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999998e284

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 97.8

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

   5. Simplified 97.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+285}:\\ \;\;\;\;\frac{y \cdot z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \mathsf{fma}\left(y, \frac{z}{t\_1 \cdot \left(x + 1\right)}, 1\right)\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -100000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.0001:\\ \;\;\;\;x + \frac{y - \frac{x}{z}}{t}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (fma y (/ z (* t_1 (+ x 1.0))) 1.0))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_3 -100000.0)
     t_2
     (if (<= t_3 0.0001)
       (+ x (/ (- y (/ x z)) t))
       (if (<= t_3 INFINITY) t_2 (/ (+ x (/ y t)) (+ x 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = fma(y, (z / (t_1 * (x + 1.0))), 1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -100000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.0001) {
		tmp = x + ((y - (x / z)) / t);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = fma(y, Float64(z / Float64(t_1 * Float64(x + 1.0))), 1.0)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -100000.0)
		tmp = t_2;
	elseif (t_3 <= 0.0001)
		tmp = Float64(x + Float64(Float64(y - Float64(x / z)) / t));
	elseif (t_3 <= Inf)
		tmp = t_2;
	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[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z / N[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -100000.0], t$95$2, If[LessEqual[t$95$3, 0.0001], N[(x + N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, 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))) < -1e5 or 1.00000000000000005e-4 < (/.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 91.7%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 85.6

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

   5. Simplified 85.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 92.8

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

   8. Simplified 92.8%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 98.4%

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

   if -1e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 93.9%

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

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

      12. /-lowering-/.f64 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 96.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -100000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}, 1\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;x + \frac{y - \frac{x}{z}}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := z \cdot t - x\\ t_3 := \mathsf{fma}\left(y, \frac{z}{t\_2 \cdot \left(x + 1\right)}, 1\right)\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -100000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 0.0001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* z t) x))
        (t_3 (fma y (/ z (* t_2 (+ x 1.0))) 1.0))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_4 -100000.0)
     t_3
     (if (<= t_4 0.0001) t_1 (if (<= t_4 INFINITY) 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 = (z * t) - x;
	double t_3 = fma(y, (z / (t_2 * (x + 1.0))), 1.0);
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -100000.0) {
		tmp = t_3;
	} else if (t_4 <= 0.0001) {
		tmp = t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		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(z * t) - x)
	t_3 = fma(y, Float64(z / Float64(t_2 * Float64(x + 1.0))), 1.0)
	t_4 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -100000.0)
		tmp = t_3;
	elseif (t_4 <= 0.0001)
		tmp = t_1;
	elseif (t_4 <= Inf)
		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[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(z / N[(t$95$2 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -100000.0], t$95$3, If[LessEqual[t$95$4, 0.0001], t$95$1, If[LessEqual[t$95$4, Infinity], t$95$3, 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))) < -1e5 or 1.00000000000000005e-4 < (/.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 91.7%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 85.6

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

   5. Simplified 85.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 92.8

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

   8. Simplified 92.8%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 98.4%

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

   if -1e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 77.1%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 89.9

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

   5. Simplified 89.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -100000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}, 1\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_2 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_2 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t_1 -1e+195)
     (fma (- 0.0 y) (/ z (fma x x x)) 1.0)
     (if (<= t_1 0.0001) t_2 (if (<= t_1 1.002) 1.0 t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_2 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e+195) {
		tmp = fma((0.0 - y), (z / fma(x, x, x)), 1.0);
	} else if (t_1 <= 0.0001) {
		tmp = t_2;
	} else if (t_1 <= 1.002) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+195)
		tmp = fma(Float64(0.0 - y), Float64(z / fma(x, x, x)), 1.0);
	elseif (t_1 <= 0.0001)
		tmp = t_2;
	elseif (t_1 <= 1.002)
		tmp = 1.0;
	else
		tmp = t_2;
	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[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+195], N[(N[(0.0 - y), $MachinePrecision] * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], t$95$2, If[LessEqual[t$95$1, 1.002], 1.0, t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.99999999999999977e194

   1. Initial program 48.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out-- N/A

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

      9. distribute-lft-out-- N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 67.2%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. div-sub N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. div-sub N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 72.7

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

   8. Simplified 72.7%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. associate-/l* N/A

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

       4. distribute-lft-neg-in N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{0 - y}, \frac{z}{x \cdot \left(1 + x\right)}, 1\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{0 - y}, \frac{z}{x \cdot \left(1 + x\right)}, 1\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(0 - y, \color{blue}{\frac{z}{x \cdot \left(1 + x\right)}}, 1\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{x \cdot \color{blue}{\left(x + 1\right)}}, 1\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{\color{blue}{x \cdot x + x \cdot 1}}, 1\right) \]

       11. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{x \cdot x + \color{blue}{x}}, 1\right) \]

       12. accelerator-lowering-fma.f64 83.0

           \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}}, 1\right) \]

   11. Simplified 83.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)} \]

   if -9.99999999999999977e194 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4 or 1.002 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.5%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 73.0

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

   5. Simplified 73.0%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.002

   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. Simplified 99.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -2e+75)
     (fma (- 0.0 y) (/ z (fma x x x)) 1.0)
     (if (<= t_1 0.0001)
       (+ x (/ y t))
       (if (<= t_1 50000.0) 1.0 (/ (/ 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 <= -2e+75) {
		tmp = fma((0.0 - y), (z / fma(x, x, x)), 1.0);
	} else if (t_1 <= 0.0001) {
		tmp = x + (y / t);
	} else if (t_1 <= 50000.0) {
		tmp = 1.0;
	} else {
		tmp = (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 <= -2e+75)
		tmp = fma(Float64(0.0 - y), Float64(z / fma(x, x, x)), 1.0);
	elseif (t_1 <= 0.0001)
		tmp = Float64(x + Float64(y / t));
	elseif (t_1 <= 50000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+75], N[(N[(0.0 - y), $MachinePrecision] * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000.0], 1.0, N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999985e75

   1. Initial program 60.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out-- N/A

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

      9. distribute-lft-out-- N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 61.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. div-sub N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. div-sub N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 66.1

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

   8. Simplified 66.1%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. associate-/l* N/A

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

       4. distribute-lft-neg-in N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{0 - y}, \frac{z}{x \cdot \left(1 + x\right)}, 1\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{0 - y}, \frac{z}{x \cdot \left(1 + x\right)}, 1\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(0 - y, \color{blue}{\frac{z}{x \cdot \left(1 + x\right)}}, 1\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{x \cdot \color{blue}{\left(x + 1\right)}}, 1\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{\color{blue}{x \cdot x + x \cdot 1}}, 1\right) \]

       11. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{x \cdot x + \color{blue}{x}}, 1\right) \]

       12. accelerator-lowering-fma.f64 74.0

           \[\leadsto \mathsf{fma}\left(0 - y, \frac{z}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}}, 1\right) \]

   11. Simplified 74.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)} \]

   if -1.99999999999999985e75 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 94.3%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 82.5%

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

      1. /-rgt-identity N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 82.5

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

   10. Applied egg-rr 82.5%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5e4

   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. Simplified 97.7%

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

   if 5e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 54.2

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

   5. Simplified 54.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 50000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -2e+75)
     (fma z (/ (- t y) (fma x x x)) 1.0)
     (if (<= t_1 0.0001)
       (+ x (/ y t))
       (if (<= t_1 50000.0) 1.0 (/ (/ 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 <= -2e+75) {
		tmp = fma(z, ((t - y) / fma(x, x, x)), 1.0);
	} else if (t_1 <= 0.0001) {
		tmp = x + (y / t);
	} else if (t_1 <= 50000.0) {
		tmp = 1.0;
	} else {
		tmp = (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 <= -2e+75)
		tmp = fma(z, Float64(Float64(t - y) / fma(x, x, x)), 1.0);
	elseif (t_1 <= 0.0001)
		tmp = Float64(x + Float64(y / t));
	elseif (t_1 <= 50000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y / t) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+75], N[(z * N[(N[(t - y), $MachinePrecision] / N[(x * x + x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000.0], 1.0, N[(N[(y / t), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999985e75

   1. Initial program 60.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out-- N/A

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

      9. distribute-lft-out-- N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 61.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. div-sub N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. div-sub N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 66.1

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

   8. Simplified 66.1%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. sub-neg N/A

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

       4. div-sub N/A

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

       5. /-lowering-/.f64 N/A

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

       6. --lowering--.f64 N/A

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

       7. +-commutative N/A

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

       8. distribute-lft-in N/A

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

       9. *-rgt-identity N/A

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

       10. accelerator-lowering-fma.f64 61.6

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

   11. Simplified 61.6%

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

   if -1.99999999999999985e75 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 94.3%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 82.5%

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

      1. /-rgt-identity N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 82.5

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

   10. Applied egg-rr 82.5%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5e4

   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. Simplified 97.7%

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

   if 5e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 54.2

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

   5. Simplified 54.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 50000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -5e-48)
     (/ y t)
     (if (<= t_1 0.0001) x (if (<= t_1 50000.0) 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 <= -5e-48) {
		tmp = y / t;
	} else if (t_1 <= 0.0001) {
		tmp = x;
	} else if (t_1 <= 50000.0) {
		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 <= (-5d-48)) then
        tmp = y / t
    else if (t_1 <= 0.0001d0) then
        tmp = x
    else if (t_1 <= 50000.0d0) 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 <= -5e-48) {
		tmp = y / t;
	} else if (t_1 <= 0.0001) {
		tmp = x;
	} else if (t_1 <= 50000.0) {
		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 <= -5e-48:
		tmp = y / t
	elif t_1 <= 0.0001:
		tmp = x
	elif t_1 <= 50000.0:
		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 <= -5e-48)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.0001)
		tmp = x;
	elseif (t_1 <= 50000.0)
		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 <= -5e-48)
		tmp = y / t;
	elseif (t_1 <= 0.0001)
		tmp = x;
	elseif (t_1 <= 50000.0)
		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, -5e-48], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], x, If[LessEqual[t$95$1, 50000.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999999e-48 or 5e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 47.9

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

   5. Simplified 47.9%

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

   if -4.9999999999999999e-48 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 93.5%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 86.8

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

   5. Simplified 86.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 84.2%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 66.3%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5e4

   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. Simplified 97.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 50000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 1.0 (if (<= t_1 0.0001) 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 = 1.0;
	} else if (t_1 <= 0.0001) {
		tmp = x;
	} else {
		tmp = 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 = 1.0;
	} else if (t_1 <= 0.0001) {
		tmp = x;
	} else {
		tmp = 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 = 1.0
	elif t_1 <= 0.0001:
		tmp = x
	else:
		tmp = 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 = 1.0;
	elseif (t_1 <= 0.0001)
		tmp = x;
	else
		tmp = 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 = 1.0;
	elseif (t_1 <= 0.0001)
		tmp = x;
	else
		tmp = 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)], 1.0, If[LessEqual[t$95$1, 0.0001], x, 1.0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 86.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 69.1%

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

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

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 76.3

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

   5. Simplified 76.3%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 73.6%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 49.3%

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

4. Final simplification 64.5%

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

Alternative 11: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ \mathsf{fma}\left(y, \frac{z}{t\_1 \cdot \left(x + 1\right)}, \frac{x}{x + 1}\right) + \frac{x}{t\_1 \cdot \left(-1 - x\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)))
   (+
    (fma y (/ z (* t_1 (+ x 1.0))) (/ x (+ x 1.0)))
    (/ x (* t_1 (- -1.0 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	return fma(y, (z / (t_1 * (x + 1.0))), (x / (x + 1.0))) + (x / (t_1 * (-1.0 - x)));
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	return Float64(fma(y, Float64(z / Float64(t_1 * Float64(x + 1.0))), Float64(x / Float64(x + 1.0))) + Float64(x / Float64(t_1 * Float64(-1.0 - x))))
end

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. --lowering--.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

   8. --lowering--.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. +-commutative N/A

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

   14. +-lowering-+.f64 N/A

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

   15. /-lowering-/.f64 N/A

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

   16. *-commutative N/A

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

5. Simplified 96.5%

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

6. Final simplification 96.5%

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

Alternative 12: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3900000:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{z \cdot \left(t - y\right)}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 29000:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3900000.0)
   (+ 1.0 (/ -1.0 x))
   (if (<= x -1.8e-53)
     (/ (* z (- t y)) (fma x x x))
     (if (<= x 29000.0) (+ x (/ y t)) 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3900000.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (x <= -1.8e-53) {
		tmp = (z * (t - y)) / fma(x, x, x);
	} else if (x <= 29000.0) {
		tmp = x + (y / t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3900000.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (x <= -1.8e-53)
		tmp = Float64(Float64(z * Float64(t - y)) / fma(x, x, x));
	elseif (x <= 29000.0)
		tmp = Float64(x + Float64(y / t));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3900000.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-53], N[(N[(z * N[(t - y), $MachinePrecision]), $MachinePrecision] / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 29000.0], N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.9e6

   1. Initial program 86.1%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 86.1

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

   5. Simplified 86.1%

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

   6. Taylor expanded in x around inf

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 92.0

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

   8. Simplified 92.0%

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

   if -3.9e6 < x < -1.7999999999999999e-53

   1. Initial program 92.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out-- N/A

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

      9. distribute-lft-out-- N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 60.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. div-sub N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. div-sub N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 60.8

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

   8. Simplified 60.8%

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

   9. Taylor expanded in z around inf

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

       1. div-sub N/A

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

       2. associate-/l* N/A

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

       3. /-lowering-/.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. --lowering--.f64 N/A

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

       6. +-commutative N/A

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

       7. distribute-lft-in N/A

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

       8. *-rgt-identity N/A

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

       9. accelerator-lowering-fma.f64 54.3

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

   11. Simplified 54.3%

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

   if -1.7999999999999999e-53 < x < 29000

   1. Initial program 90.1%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 70.4%

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

      1. /-rgt-identity N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 70.4

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

   10. Applied egg-rr 70.4%

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

   if 29000 < x

   1. Initial program 87.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 91.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3900000:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{z \cdot \left(t - y\right)}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 29000:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.063:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 115000:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.063) (/ x (+ x 1.0)) (if (<= x 115000.0) (+ x (/ y t)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.063) {
		tmp = x / (x + 1.0);
	} else if (x <= 115000.0) {
		tmp = x + (y / t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-0.063d0)) then
        tmp = x / (x + 1.0d0)
    else if (x <= 115000.0d0) then
        tmp = x + (y / t)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.063) {
		tmp = x / (x + 1.0);
	} else if (x <= 115000.0) {
		tmp = x + (y / t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.063:
		tmp = x / (x + 1.0)
	elif x <= 115000.0:
		tmp = x + (y / t)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.063)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 115000.0)
		tmp = Float64(x + Float64(y / t));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -0.063)
		tmp = x / (x + 1.0);
	elseif (x <= 115000.0)
		tmp = x + (y / t);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.063], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 115000.0], N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.063

   1. Initial program 86.5%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 89.3

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

   5. Simplified 89.3%

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

   if -0.063 < x < 115000

   1. Initial program 90.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 68.0

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

   5. Simplified 68.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 66.1%

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

      1. /-rgt-identity N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 66.1

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

   10. Applied egg-rr 66.1%

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

   if 115000 < x

   1. Initial program 87.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 91.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.063:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 115000:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 29000:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.0) (+ 1.0 (/ -1.0 x)) (if (<= x 29000.0) (+ x (/ y t)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (x <= 29000.0) {
		tmp = x + (y / t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (x <= 29000.0d0) then
        tmp = x + (y / t)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (x <= 29000.0) {
		tmp = x + (y / t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0 + (-1.0 / x)
	elif x <= 29000.0:
		tmp = x + (y / t)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (x <= 29000.0)
		tmp = Float64(x + Float64(y / t));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0 + (-1.0 / x);
	elseif (x <= 29000.0)
		tmp = x + (y / t);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 29000.0], N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 86.5%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 86.5

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

   5. Simplified 86.5%

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

   6. Taylor expanded in x around inf

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 89.3

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

   8. Simplified 89.3%

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

   if -1 < x < 29000

   1. Initial program 90.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 68.0

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

   5. Simplified 68.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 66.1%

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

      1. /-rgt-identity N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 66.1

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

   10. Applied egg-rr 66.1%

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

   if 29000 < x

   1. Initial program 87.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 91.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 29000:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.62:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 29000:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.62) 1.0 (if (<= x 29000.0) (+ x (/ y t)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.62) {
		tmp = 1.0;
	} else if (x <= 29000.0) {
		tmp = x + (y / t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-0.62d0)) then
        tmp = 1.0d0
    else if (x <= 29000.0d0) then
        tmp = x + (y / t)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.62) {
		tmp = 1.0;
	} else if (x <= 29000.0) {
		tmp = x + (y / t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.62:
		tmp = 1.0
	elif x <= 29000.0:
		tmp = x + (y / t)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.62)
		tmp = 1.0;
	elseif (x <= 29000.0)
		tmp = Float64(x + Float64(y / t));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -0.62)
		tmp = 1.0;
	elseif (x <= 29000.0)
		tmp = x + (y / t);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.62], 1.0, If[LessEqual[x, 29000.0], N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -0.619999999999999996 or 29000 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 90.3%

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

   if -0.619999999999999996 < x < 29000

   1. Initial program 90.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 68.0

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

   5. Simplified 68.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 66.1%

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

      1. /-rgt-identity N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 66.1

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

   10. Applied egg-rr 66.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.62:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 29000:\\ \;\;\;\;x + \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.8% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 54.1%

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

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024195 
(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)))