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

Percentage Accurate: 89.8% → 94.8%

Time: 14.5s

Alternatives: 21

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1} + \frac{\frac{y}{x + 1} - \frac{x}{\mathsf{fma}\left(x, z, z\right)}}{t}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x (+ x 1.0)) (/ (- (/ y (+ x 1.0)) (/ x (fma x z z))) t))))
   (if (<= z -6.5e+173)
     t_1
     (if (<= z 3.9e+112)
       (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / (x + 1.0)) + (((y / (x + 1.0)) - (x / fma(x, z, z))) / t);
	double tmp;
	if (z <= -6.5e+173) {
		tmp = t_1;
	} else if (z <= 3.9e+112) {
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(Float64(y / Float64(x + 1.0)) - Float64(x / fma(x, z, z))) / t))
	tmp = 0.0
	if (z <= -6.5e+173)
		tmp = t_1;
	elseif (z <= 3.9e+112)
		tmp = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / 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[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(x * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e+173], t$95$1, If[LessEqual[z, 3.9e+112], N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4999999999999997e173 or 3.89999999999999968e112 < z

   1. Initial program 66.1%

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

   3. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

   5. Simplified 88.6%

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

   if -6.4999999999999997e173 < z < 3.89999999999999968e112

   1. Initial program 99.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+173}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{\frac{y}{x + 1} - \frac{x}{\mathsf{fma}\left(x, z, z\right)}}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{\frac{y}{x + 1} - \frac{x}{\mathsf{fma}\left(x, z, z\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{z \cdot y - x}{t\_1}}{x + 1}\\ t_3 := \frac{z \cdot y}{\left(x + 1\right) \cdot t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;t\_2 \leq -1000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(y, z, 0 - x\right)}{z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(z, t, 0 - x\right)}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{+205}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (/ (+ x (/ (- (* z y) x) t_1)) (+ x 1.0)))
        (t_3 (/ (* z y) (* (+ x 1.0) t_1))))
   (if (<= t_2 -2e+254)
     (/ y (fma x t t))
     (if (<= t_2 -1000000000.0)
       t_3
       (if (<= t_2 2e-63)
         (/ (+ x (/ (fma y z (- 0.0 x)) (* z t))) (+ x 1.0))
         (if (<= t_2 2.0)
           (/ (- x (/ x (fma z t (- 0.0 x)))) (+ x 1.0))
           (if (<= t_2 1e+205) t_3 (/ (+ 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 = (x + (((z * y) - x) / t_1)) / (x + 1.0);
	double t_3 = (z * y) / ((x + 1.0) * t_1);
	double tmp;
	if (t_2 <= -2e+254) {
		tmp = y / fma(x, t, t);
	} else if (t_2 <= -1000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 2e-63) {
		tmp = (x + (fma(y, z, (0.0 - x)) / (z * t))) / (x + 1.0);
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / fma(z, t, (0.0 - x)))) / (x + 1.0);
	} else if (t_2 <= 1e+205) {
		tmp = t_3;
	} 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 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / t_1)) / Float64(x + 1.0))
	t_3 = Float64(Float64(z * y) / Float64(Float64(x + 1.0) * t_1))
	tmp = 0.0
	if (t_2 <= -2e+254)
		tmp = Float64(y / fma(x, t, t));
	elseif (t_2 <= -1000000000.0)
		tmp = t_3;
	elseif (t_2 <= 2e-63)
		tmp = Float64(Float64(x + Float64(fma(y, z, Float64(0.0 - x)) / Float64(z * t))) / Float64(x + 1.0));
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(z, t, Float64(0.0 - x)))) / Float64(x + 1.0));
	elseif (t_2 <= 1e+205)
		tmp = t_3;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * y), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+254], N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1000000000.0], t$95$3, If[LessEqual[t$95$2, 2e-63], N[(N[(x + N[(N[(y * z + N[(0.0 - x), $MachinePrecision]), $MachinePrecision] / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / N[(z * t + N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+205], t$95$3, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.9999999999999999e254

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 61.7

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

   5. Simplified 61.7%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 74.1

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

   8. Simplified 74.1%

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

   if -1.9999999999999999e254 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e9 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000002e205

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. /-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 96.1

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

   5. Simplified 96.1%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e-63

   1. Initial program 93.6%

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

   3. Taylor expanded in t around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

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

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

      8. *-lowering-*.f64 93.1

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

   5. Simplified 93.1%

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

   if 2.00000000000000013e-63 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 99.1

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

   5. Simplified 99.1%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. sub0-neg N/A

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

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

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

      5. --lowering--.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if 1.00000000000000002e205 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 78.3

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

   5. Simplified 78.3%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -1000000000:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(y, z, 0 - x\right)}{z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(z, t, 0 - x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 10^{+205}:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot y - x\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{t\_1}{t\_2}}{x + 1}\\ t_4 := \frac{z \cdot y}{\left(x + 1\right) \cdot t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;t\_3 \leq -1000000000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{x + \frac{t\_1}{z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(z, t, 0 - x\right)}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 10^{+205}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z y) x))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ t_1 t_2)) (+ x 1.0)))
        (t_4 (/ (* z y) (* (+ x 1.0) t_2))))
   (if (<= t_3 -2e+254)
     (/ y (fma x t t))
     (if (<= t_3 -1000000000.0)
       t_4
       (if (<= t_3 2e-63)
         (/ (+ x (/ t_1 (* z t))) (+ x 1.0))
         (if (<= t_3 2.0)
           (/ (- x (/ x (fma z t (- 0.0 x)))) (+ x 1.0))
           (if (<= t_3 1e+205) t_4 (/ (+ x (/ y t)) (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) - x;
	double t_2 = (z * t) - x;
	double t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	double t_4 = (z * y) / ((x + 1.0) * t_2);
	double tmp;
	if (t_3 <= -2e+254) {
		tmp = y / fma(x, t, t);
	} else if (t_3 <= -1000000000.0) {
		tmp = t_4;
	} else if (t_3 <= 2e-63) {
		tmp = (x + (t_1 / (z * t))) / (x + 1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / fma(z, t, (0.0 - x)))) / (x + 1.0);
	} else if (t_3 <= 1e+205) {
		tmp = t_4;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) - x)
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(t_1 / t_2)) / Float64(x + 1.0))
	t_4 = Float64(Float64(z * y) / Float64(Float64(x + 1.0) * t_2))
	tmp = 0.0
	if (t_3 <= -2e+254)
		tmp = Float64(y / fma(x, t, t));
	elseif (t_3 <= -1000000000.0)
		tmp = t_4;
	elseif (t_3 <= 2e-63)
		tmp = Float64(Float64(x + Float64(t_1 / Float64(z * t))) / Float64(x + 1.0));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(z, t, Float64(0.0 - x)))) / Float64(x + 1.0));
	elseif (t_3 <= 1e+205)
		tmp = t_4;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z * y), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+254], N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1000000000.0], t$95$4, If[LessEqual[t$95$3, 2e-63], N[(N[(x + N[(t$95$1 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / N[(z * t + N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+205], t$95$4, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.9999999999999999e254

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 61.7

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

   5. Simplified 61.7%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 74.1

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

   8. Simplified 74.1%

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

   if -1.9999999999999999e254 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e9 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000002e205

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. /-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 96.1

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

   5. Simplified 96.1%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e-63

   1. Initial program 93.6%

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

   3. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 92.9

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

   5. Simplified 92.9%

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

   if 2.00000000000000013e-63 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 99.1

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

   5. Simplified 99.1%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. sub0-neg N/A

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

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

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

      5. --lowering--.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if 1.00000000000000002e205 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 78.3

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

   5. Simplified 78.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -1000000000:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(z, t, 0 - x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 10^{+205}:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{z \cdot y - x}{t\_1}}{x + 1}\\ t_3 := \frac{z \cdot y}{\left(x + 1\right) \cdot t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;t\_2 \leq -1000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(z, t, 0 - x\right)}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{+205}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (/ (+ x (/ (- (* z y) x) t_1)) (+ x 1.0)))
        (t_3 (/ (* z y) (* (+ x 1.0) t_1))))
   (if (<= t_2 -2e+254)
     (/ y (fma x t t))
     (if (<= t_2 -1000000000.0)
       t_3
       (if (<= t_2 2e-63)
         (fma (- 1.0 x) (/ y t) x)
         (if (<= t_2 2.0)
           (/ (- x (/ x (fma z t (- 0.0 x)))) (+ x 1.0))
           (if (<= t_2 1e+205) t_3 (/ (+ 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 = (x + (((z * y) - x) / t_1)) / (x + 1.0);
	double t_3 = (z * y) / ((x + 1.0) * t_1);
	double tmp;
	if (t_2 <= -2e+254) {
		tmp = y / fma(x, t, t);
	} else if (t_2 <= -1000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 2e-63) {
		tmp = fma((1.0 - x), (y / t), x);
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / fma(z, t, (0.0 - x)))) / (x + 1.0);
	} else if (t_2 <= 1e+205) {
		tmp = t_3;
	} 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 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / t_1)) / Float64(x + 1.0))
	t_3 = Float64(Float64(z * y) / Float64(Float64(x + 1.0) * t_1))
	tmp = 0.0
	if (t_2 <= -2e+254)
		tmp = Float64(y / fma(x, t, t));
	elseif (t_2 <= -1000000000.0)
		tmp = t_3;
	elseif (t_2 <= 2e-63)
		tmp = fma(Float64(1.0 - x), Float64(y / t), x);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(z, t, Float64(0.0 - x)))) / Float64(x + 1.0));
	elseif (t_2 <= 1e+205)
		tmp = t_3;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * y), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+254], N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1000000000.0], t$95$3, If[LessEqual[t$95$2, 2e-63], N[(N[(1.0 - x), $MachinePrecision] * N[(y / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / N[(z * t + N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+205], t$95$3, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.9999999999999999e254

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 61.7

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

   5. Simplified 61.7%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 74.1

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

   8. Simplified 74.1%

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

   if -1.9999999999999999e254 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e9 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000002e205

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. /-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 96.1

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

   5. Simplified 96.1%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e-63

   1. Initial program 93.6%

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

   3. Taylor expanded in z around inf

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

      1. /-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 85.7

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

   5. Simplified 85.7%

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

   6. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 85.7

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

   8. Simplified 85.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. mul-1-neg N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. associate-/l* N/A

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

       9. mul-1-neg N/A

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

       10. *-rgt-identity N/A

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

       11. associate-+r+ N/A

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

   11. Simplified 85.7%

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

   if 2.00000000000000013e-63 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 99.1

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

   5. Simplified 99.1%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. sub0-neg N/A

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

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

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

      5. --lowering--.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if 1.00000000000000002e205 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 78.3

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

   5. Simplified 78.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -1000000000:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(z, t, 0 - x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 10^{+205}:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{z \cdot y - x}{t\_1}}{x + 1}\\ t_3 := \frac{z \cdot y}{\left(x + 1\right) \cdot t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;t\_2 \leq -1000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{+205}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (/ (+ x (/ (- (* z y) x) t_1)) (+ x 1.0)))
        (t_3 (/ (* z y) (* (+ x 1.0) t_1))))
   (if (<= t_2 -2e+254)
     (/ y (fma x t t))
     (if (<= t_2 -1000000000.0)
       t_3
       (if (<= t_2 2e-63)
         (fma (- 1.0 x) (/ y t) x)
         (if (<= t_2 2.0)
           (/ (+ x (/ x (- x (* z t)))) (+ x 1.0))
           (if (<= t_2 1e+205) t_3 (/ (+ 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 = (x + (((z * y) - x) / t_1)) / (x + 1.0);
	double t_3 = (z * y) / ((x + 1.0) * t_1);
	double tmp;
	if (t_2 <= -2e+254) {
		tmp = y / fma(x, t, t);
	} else if (t_2 <= -1000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 2e-63) {
		tmp = fma((1.0 - x), (y / t), x);
	} else if (t_2 <= 2.0) {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	} else if (t_2 <= 1e+205) {
		tmp = t_3;
	} 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 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / t_1)) / Float64(x + 1.0))
	t_3 = Float64(Float64(z * y) / Float64(Float64(x + 1.0) * t_1))
	tmp = 0.0
	if (t_2 <= -2e+254)
		tmp = Float64(y / fma(x, t, t));
	elseif (t_2 <= -1000000000.0)
		tmp = t_3;
	elseif (t_2 <= 2e-63)
		tmp = fma(Float64(1.0 - x), Float64(y / t), x);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x + Float64(x / Float64(x - Float64(z * t)))) / Float64(x + 1.0));
	elseif (t_2 <= 1e+205)
		tmp = t_3;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * y), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+254], N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1000000000.0], t$95$3, If[LessEqual[t$95$2, 2e-63], N[(N[(1.0 - x), $MachinePrecision] * N[(y / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(x + N[(x / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+205], t$95$3, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.9999999999999999e254

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 61.7

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

   5. Simplified 61.7%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 74.1

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

   8. Simplified 74.1%

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

   if -1.9999999999999999e254 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e9 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000002e205

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. /-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 96.1

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

   5. Simplified 96.1%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000013e-63

   1. Initial program 93.6%

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

   3. Taylor expanded in z around inf

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

      1. /-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 85.7

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

   5. Simplified 85.7%

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

   6. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 85.7

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

   8. Simplified 85.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. mul-1-neg N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. associate-/l* N/A

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

       9. mul-1-neg N/A

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

       10. *-rgt-identity N/A

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

       11. associate-+r+ N/A

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

   11. Simplified 85.7%

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

   if 2.00000000000000013e-63 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 99.1

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

   5. Simplified 99.1%

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

   if 1.00000000000000002e205 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 78.3

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

   5. Simplified 78.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -1000000000:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 10^{+205}:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{z \cdot y - x}{t\_1}}{x + 1}\\ t_3 := \frac{z \cdot y}{\left(x + 1\right) \cdot t\_1}\\ t_4 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;t\_2 \leq -1000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.9999999999999853:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+205}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (/ (+ x (/ (- (* z y) x) t_1)) (+ x 1.0)))
        (t_3 (/ (* z y) (* (+ x 1.0) t_1)))
        (t_4 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t_2 -2e+254)
     (/ y (fma x t t))
     (if (<= t_2 -1000000000.0)
       t_3
       (if (<= t_2 0.9999999999999853)
         t_4
         (if (<= t_2 20.0)
           (fma (- 0.0 y) (/ z (fma x x x)) 1.0)
           (if (<= t_2 1e+205) t_3 t_4)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((z * y) - x) / t_1)) / (x + 1.0);
	double t_3 = (z * y) / ((x + 1.0) * t_1);
	double t_4 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+254) {
		tmp = y / fma(x, t, t);
	} else if (t_2 <= -1000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 0.9999999999999853) {
		tmp = t_4;
	} else if (t_2 <= 20.0) {
		tmp = fma((0.0 - y), (z / fma(x, x, x)), 1.0);
	} else if (t_2 <= 1e+205) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / t_1)) / Float64(x + 1.0))
	t_3 = Float64(Float64(z * y) / Float64(Float64(x + 1.0) * t_1))
	t_4 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e+254)
		tmp = Float64(y / fma(x, t, t));
	elseif (t_2 <= -1000000000.0)
		tmp = t_3;
	elseif (t_2 <= 0.9999999999999853)
		tmp = t_4;
	elseif (t_2 <= 20.0)
		tmp = fma(Float64(0.0 - y), Float64(z / fma(x, x, x)), 1.0);
	elseif (t_2 <= 1e+205)
		tmp = t_3;
	else
		tmp = t_4;
	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[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * y), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+254], N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1000000000.0], t$95$3, If[LessEqual[t$95$2, 0.9999999999999853], t$95$4, If[LessEqual[t$95$2, 20.0], N[(N[(0.0 - y), $MachinePrecision] * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+205], t$95$3, t$95$4]]]]]]]]]

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))) < -1.9999999999999999e254

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 61.7

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

   5. Simplified 61.7%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 74.1

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

   8. Simplified 74.1%

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

   if -1.9999999999999999e254 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e9 or 20 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000002e205

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. /-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%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999999999998535 or 1.00000000000000002e205 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x + \frac{y}{t}}{1 + x}} \]
   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 80.7

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

   5. Simplified 80.7%

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

   if 0.99999999999998535 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 20

   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 t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

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

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

      10. /-lowering-/.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. 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. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. accelerator-lowering-fma.f64 99.5

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

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -1000000000:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 0.9999999999999853:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 10^{+205}:\\ \;\;\;\;\frac{z \cdot y}{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -1e+100)
     (/ y (fma x t t))
     (if (<= t_1 0.0001)
       (fma (- 1.0 x) (/ y t) x)
       (if (<= t_1 20.0) 1.0 (/ (/ y (+ x 1.0)) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e+100) {
		tmp = y / fma(x, t, t);
	} else if (t_1 <= 0.0001) {
		tmp = fma((1.0 - x), (y / t), x);
	} else if (t_1 <= 20.0) {
		tmp = 1.0;
	} else {
		tmp = (y / (x + 1.0)) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+100)
		tmp = Float64(y / fma(x, t, t));
	elseif (t_1 <= 0.0001)
		tmp = fma(Float64(1.0 - x), Float64(y / t), x);
	elseif (t_1 <= 20.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y / Float64(x + 1.0)) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+100], N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[(1.0 - x), $MachinePrecision] * N[(y / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 20.0], 1.0, N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.00000000000000002e100

   1. Initial program 63.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 66.4

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

   5. Simplified 66.4%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 73.1

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

   8. Simplified 73.1%

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

   if -1.00000000000000002e100 < (/.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 95.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 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 t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 76.2

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

   8. Simplified 76.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. mul-1-neg N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. associate-/l* N/A

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

       9. mul-1-neg N/A

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

       10. *-rgt-identity N/A

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

       11. associate-+r+ N/A

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

   11. Simplified 75.3%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 20

   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 97.3%

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

   if 20 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 59.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 70.8

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

   5. Simplified 70.8%

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

   6. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 70.9

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

   8. Simplified 70.9%

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

   9. Taylor expanded in t around 0

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

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

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

       2. +-lowering-+.f64 57.8

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

   11. Simplified 57.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -1e+100)
     (/ y (fma x t t))
     (if (<= t_1 0.0001)
       (fma (- 1.0 x) (/ y t) x)
       (if (<= t_1 20.0) 1.0 (/ (/ y t) (+ x 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e+100) {
		tmp = y / fma(x, t, t);
	} else if (t_1 <= 0.0001) {
		tmp = fma((1.0 - x), (y / t), x);
	} else if (t_1 <= 20.0) {
		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(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+100)
		tmp = Float64(y / fma(x, t, t));
	elseif (t_1 <= 0.0001)
		tmp = fma(Float64(1.0 - x), Float64(y / t), x);
	elseif (t_1 <= 20.0)
		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[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+100], N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[(1.0 - x), $MachinePrecision] * N[(y / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 20.0], 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))) < -1.00000000000000002e100

   1. Initial program 63.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 66.4

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

   5. Simplified 66.4%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 73.1

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

   8. Simplified 73.1%

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

   if -1.00000000000000002e100 < (/.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 95.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 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 t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 76.2

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

   8. Simplified 76.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. mul-1-neg N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. associate-/l* N/A

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

       9. mul-1-neg N/A

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

       10. *-rgt-identity N/A

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

       11. associate-+r+ N/A

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

   11. Simplified 75.3%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 20

   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 97.3%

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

   if 20 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 59.5%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 57.6

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

   5. Simplified 57.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -1e+100)
     t_1
     (if (<= t_2 0.0001)
       (fma (- 1.0 x) (/ y t) x)
       (if (<= t_2 20.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -1e+100) {
		tmp = t_1;
	} else if (t_2 <= 0.0001) {
		tmp = fma((1.0 - x), (y / t), x);
	} else if (t_2 <= 20.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1e+100)
		tmp = t_1;
	elseif (t_2 <= 0.0001)
		tmp = fma(Float64(1.0 - x), Float64(y / t), x);
	elseif (t_2 <= 20.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+100], t$95$1, If[LessEqual[t$95$2, 0.0001], N[(N[(1.0 - x), $MachinePrecision] * N[(y / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 20.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.00000000000000002e100 or 20 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 60.9%

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

   3. Taylor expanded in z around inf

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

      1. /-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 69.0

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

   5. Simplified 69.0%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 63.9

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

   8. Simplified 63.9%

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

   if -1.00000000000000002e100 < (/.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 95.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 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 t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 76.2

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

   8. Simplified 76.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. mul-1-neg N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. associate-/l* N/A

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

       9. mul-1-neg N/A

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

       10. *-rgt-identity N/A

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

       11. associate-+r+ N/A

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

   11. Simplified 75.3%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 20

   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 97.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.0001:\\ \;\;\;\;x - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -1e-46)
     t_1
     (if (<= t_2 0.0001) (- x (/ x (* z t))) (if (<= t_2 20.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -1e-46) {
		tmp = t_1;
	} else if (t_2 <= 0.0001) {
		tmp = x - (x / (z * t));
	} else if (t_2 <= 20.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1e-46)
		tmp = t_1;
	elseif (t_2 <= 0.0001)
		tmp = Float64(x - Float64(x / Float64(z * t)));
	elseif (t_2 <= 20.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-46], t$95$1, If[LessEqual[t$95$2, 0.0001], N[(x - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.00000000000000002e-46 or 20 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 66.1

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

   5. Simplified 66.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 61.1

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

   8. Simplified 61.1%

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

   if -1.00000000000000002e-46 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 68.4

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

   5. Simplified 68.4%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

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

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

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

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

      12. *-lowering-*.f64 67.1

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

   8. Simplified 67.1%

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

   if 1.00000000000000005e-4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 20

   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 97.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 0.0001:\\ \;\;\;\;x - \frac{x}{z \cdot t}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.999995:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -2e-71)
     t_1
     (if (<= t_2 0.999995) (/ x (+ x 1.0)) (if (<= t_2 20.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e-71) {
		tmp = t_1;
	} else if (t_2 <= 0.999995) {
		tmp = x / (x + 1.0);
	} else if (t_2 <= 20.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e-71)
		tmp = t_1;
	elseif (t_2 <= 0.999995)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_2 <= 20.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-71], t$95$1, If[LessEqual[t$95$2, 0.999995], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.9999999999999998e-71 or 20 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.6%

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

   3. Taylor expanded in z around inf

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

      1. /-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 68.6

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

   5. Simplified 68.6%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 60.1

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

   8. Simplified 60.1%

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

   if -1.9999999999999998e-71 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999499999999997

   1. Initial program 92.0%

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

   3. Taylor expanded in t around inf

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

      1. /-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 60.5

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

   5. Simplified 60.5%

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

   if 0.99999499999999997 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 20

   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 98.2%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 0.999995:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.999995:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -2e-71)
     (/ y t)
     (if (<= t_1 0.999995) (/ x (+ x 1.0)) (if (<= t_1 20.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e-71) {
		tmp = y / t;
	} else if (t_1 <= 0.999995) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 20.0) {
		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 + (((z * y) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= (-2d-71)) then
        tmp = y / t
    else if (t_1 <= 0.999995d0) then
        tmp = x / (x + 1.0d0)
    else if (t_1 <= 20.0d0) 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 + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e-71) {
		tmp = y / t;
	} else if (t_1 <= 0.999995) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 20.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -2e-71:
		tmp = y / t
	elif t_1 <= 0.999995:
		tmp = x / (x + 1.0)
	elif t_1 <= 20.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -2e-71)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.999995)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_1 <= 20.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -2e-71)
		tmp = y / t;
	elseif (t_1 <= 0.999995)
		tmp = x / (x + 1.0);
	elseif (t_1 <= 20.0)
		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[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-71], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.999995], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20.0], 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))) < -1.9999999999999998e-71 or 20 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 52.7

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

   5. Simplified 52.7%

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

   if -1.9999999999999998e-71 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999499999999997

   1. Initial program 92.0%

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

   3. Taylor expanded in t around inf

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

      1. /-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 60.5

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

   5. Simplified 60.5%

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

   if 0.99999499999999997 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 20

   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 98.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 0.999995:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+205}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -2e+254)
     (/ y (fma x t t))
     (if (<= t_1 1e+205) t_1 (/ (+ x (/ y t)) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e+254) {
		tmp = y / fma(x, t, t);
	} else if (t_1 <= 1e+205) {
		tmp = t_1;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -2e+254)
		tmp = Float64(y / fma(x, t, t));
	elseif (t_1 <= 1e+205)
		tmp = t_1;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+254], N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+205], t$95$1, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.9999999999999999e254

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 61.7

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

   5. Simplified 61.7%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 74.1

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

   8. Simplified 74.1%

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

   if -1.9999999999999999e254 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000002e205

   1. Initial program 98.5%

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

   if 1.00000000000000002e205 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 31.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 78.3

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

   5. Simplified 78.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 10^{+205}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 5e-98) (/ y t) (if (<= t_1 20.0) 1.0 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e-98) {
		tmp = y / t;
	} else if (t_1 <= 20.0) {
		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 + (((z * y) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= 5d-98) then
        tmp = y / t
    else if (t_1 <= 20.0d0) 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 + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e-98) {
		tmp = y / t;
	} else if (t_1 <= 20.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= 5e-98:
		tmp = y / t
	elif t_1 <= 20.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= 5e-98)
		tmp = Float64(y / t);
	elseif (t_1 <= 20.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= 5e-98)
		tmp = y / t;
	elseif (t_1 <= 20.0)
		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[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-98], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 20.0], 1.0, N[(y / t), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000018e-98 or 20 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 76.3%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 48.2

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

   5. Simplified 48.2%

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

   if 5.00000000000000018e-98 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 20

   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 92.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1} \leq 20:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+173}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, t\_1, \frac{y}{x + 1}\right)}{t}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+110}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, x, t\right)} + \left(t\_1 - \frac{x}{\mathsf{fma}\left(x, z, z\right) \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ x 1.0))))
   (if (<= z -4.2e+173)
     (/ (fma t t_1 (/ y (+ x 1.0))) t)
     (if (<= z 1.9e+110)
       (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))
       (+ (/ y (fma t x t)) (- t_1 (/ x (* (fma x z z) t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double tmp;
	if (z <= -4.2e+173) {
		tmp = fma(t, t_1, (y / (x + 1.0))) / t;
	} else if (z <= 1.9e+110) {
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	} else {
		tmp = (y / fma(t, x, t)) + (t_1 - (x / (fma(x, z, z) * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (z <= -4.2e+173)
		tmp = Float64(fma(t, t_1, Float64(y / Float64(x + 1.0))) / t);
	elseif (z <= 1.9e+110)
		tmp = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(y / fma(t, x, t)) + Float64(t_1 - Float64(x / Float64(fma(x, z, z) * t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e+173], N[(N[(t * t$95$1 + N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.9e+110], N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[(x / N[(N[(x * z + z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2e173

   1. Initial program 62.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x + \frac{y}{t}}{1 + x}} \]
   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 80.2

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

   5. Simplified 80.2%

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

   6. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 80.3

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

   8. Simplified 80.3%

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

   if -4.2e173 < z < 1.89999999999999994e110

   1. Initial program 99.3%

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

   if 1.89999999999999994e110 < z

   1. Initial program 68.2%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

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

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

      13. /-lowering-/.f64 N/A

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

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

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

      15. +-commutative N/A

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

      16. distribute-rgt-in N/A

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

      17. *-lft-identity N/A

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

      18. accelerator-lowering-fma.f64 93.3

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

   5. Simplified 93.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+173}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{x + 1}, \frac{y}{x + 1}\right)}{t}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+110}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, x, t\right)} + \left(\frac{x}{x + 1} - \frac{x}{\mathsf{fma}\left(x, z, z\right) \cdot t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{x + 1}, \frac{y}{x + 1}\right)}{t}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+109}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x (+ x 1.0)) (/ y (+ x 1.0))) t)))
   (if (<= z -9.5e+174)
     t_1
     (if (<= z 1.25e+109)
       (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, (x / (x + 1.0)), (y / (x + 1.0))) / t;
	double tmp;
	if (z <= -9.5e+174) {
		tmp = t_1;
	} else if (z <= 1.25e+109) {
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(t, Float64(x / Float64(x + 1.0)), Float64(y / Float64(x + 1.0))) / t)
	tmp = 0.0
	if (z <= -9.5e+174)
		tmp = t_1;
	elseif (z <= 1.25e+109)
		tmp = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -9.5e+174], t$95$1, If[LessEqual[z, 1.25e+109], N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999992e174 or 1.25e109 < z

   1. Initial program 66.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 86.3

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

   5. Simplified 86.3%

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

   6. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 88.6

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

   8. Simplified 88.6%

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

   if -9.4999999999999992e174 < z < 1.25e109

   1. Initial program 99.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{x + 1}, \frac{y}{x + 1}\right)}{t}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+109}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{x + 1}, \frac{y}{x + 1}\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(z, 0 - \frac{y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t -2.4e-99)
     t_1
     (if (<= t 1.55e-91) (fma z (- 0.0 (/ y (fma x x 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 tmp;
	if (t <= -2.4e-99) {
		tmp = t_1;
	} else if (t <= 1.55e-91) {
		tmp = fma(z, (0.0 - (y / fma(x, x, 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))
	tmp = 0.0
	if (t <= -2.4e-99)
		tmp = t_1;
	elseif (t <= 1.55e-91)
		tmp = fma(z, Float64(0.0 - Float64(y / fma(x, x, 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]}, If[LessEqual[t, -2.4e-99], t$95$1, If[LessEqual[t, 1.55e-91], N[(z * N[(0.0 - N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4e-99 or 1.5499999999999999e-91 < t

   1. Initial program 87.7%

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

   3. Taylor expanded in z around 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 87.8

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

   5. Simplified 87.8%

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

   if -2.4e-99 < t < 1.5499999999999999e-91

   1. Initial program 89.4%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

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

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

      10. /-lowering-/.f64 76.3

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

   5. Simplified 76.3%

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

   6. Taylor expanded in z around inf

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

      1. distribute-lft-in N/A

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

      2. rgt-mult-inverse N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. accelerator-lowering-fma.f64 78.6

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

   8. Simplified 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{0 - y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 78.6

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

   10. Applied egg-rr 78.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(z, 0 - \frac{y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 0 - \frac{y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -8.8e-26)
   (fma (- 0.0 y) (/ z (fma x x x)) 1.0)
   (if (<= x 2.15e-18)
     (fma (- 1.0 x) (/ y t) x)
     (fma z (- 0.0 (/ y (fma x x x))) 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -8.8e-26) {
		tmp = fma((0.0 - y), (z / fma(x, x, x)), 1.0);
	} else if (x <= 2.15e-18) {
		tmp = fma((1.0 - x), (y / t), x);
	} else {
		tmp = fma(z, (0.0 - (y / fma(x, x, x))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -8.8e-26)
		tmp = fma(Float64(0.0 - y), Float64(z / fma(x, x, x)), 1.0);
	elseif (x <= 2.15e-18)
		tmp = fma(Float64(1.0 - x), Float64(y / t), x);
	else
		tmp = fma(z, Float64(0.0 - Float64(y / fma(x, x, x))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -8.8e-26], N[(N[(0.0 - y), $MachinePrecision] * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 2.15e-18], N[(N[(1.0 - x), $MachinePrecision] * N[(y / t), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(0.0 - N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.8000000000000003e-26

   1. Initial program 83.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

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

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

      10. /-lowering-/.f64 88.0

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

   5. Simplified 88.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. 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. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. accelerator-lowering-fma.f64 90.6

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

   8. Simplified 90.6%

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

   if -8.8000000000000003e-26 < x < 2.1500000000000001e-18

   1. Initial program 89.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 72.6

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

   8. Simplified 72.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. mul-1-neg N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. associate-/l* N/A

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

       9. mul-1-neg N/A

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

       10. *-rgt-identity N/A

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

       11. associate-+r+ N/A

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

   11. Simplified 72.6%

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

   if 2.1500000000000001e-18 < x

   1. Initial program 90.6%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

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

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

      10. /-lowering-/.f64 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in z around inf

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

      1. distribute-lft-in N/A

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

      2. rgt-mult-inverse N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. accelerator-lowering-fma.f64 87.2

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

   8. Simplified 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{0 - y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 87.2

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

   10. Applied egg-rr 87.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 0 - \frac{y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, 0 - \frac{y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (- 0.0 (/ y (fma x x x))) 1.0)))
   (if (<= x -1.55e-23)
     t_1
     (if (<= x 4.8e-19) (fma (- 1.0 x) (/ y t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (0.0 - (y / fma(x, x, x))), 1.0);
	double tmp;
	if (x <= -1.55e-23) {
		tmp = t_1;
	} else if (x <= 4.8e-19) {
		tmp = fma((1.0 - x), (y / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(0.0 - Float64(y / fma(x, x, x))), 1.0)
	tmp = 0.0
	if (x <= -1.55e-23)
		tmp = t_1;
	elseif (x <= 4.8e-19)
		tmp = fma(Float64(1.0 - x), Float64(y / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(0.0 - N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.55e-23], t$95$1, If[LessEqual[x, 4.8e-19], N[(N[(1.0 - x), $MachinePrecision] * N[(y / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5499999999999999e-23 or 4.80000000000000046e-19 < x

   1. Initial program 87.3%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

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

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

      10. /-lowering-/.f64 87.6

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

   5. Simplified 87.6%

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

   6. Taylor expanded in z around inf

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

      1. distribute-lft-in N/A

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

      2. rgt-mult-inverse N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. accelerator-lowering-fma.f64 88.9

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

   8. Simplified 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{0 - y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 88.9

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

   10. Applied egg-rr 88.9%

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

   if -1.5499999999999999e-23 < x < 4.80000000000000046e-19

   1. Initial program 89.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{t \cdot x}{1 + x} + \frac{y}{1 + x}}}{t} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{t \cdot \frac{x}{1 + x}} + \frac{y}{1 + x}}{t} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t, \frac{x}{1 + x}, \frac{y}{1 + x}\right)}}{t} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \color{blue}{\frac{x}{1 + x}}, \frac{y}{1 + x}\right)}{t} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{\color{blue}{x + 1}}, \frac{y}{1 + x}\right)}{t} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{\color{blue}{x + 1}}, \frac{y}{1 + x}\right)}{t} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{x + 1}, \color{blue}{\frac{y}{1 + x}}\right)}{t} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{x + 1}, \frac{y}{\color{blue}{x + 1}}\right)}{t} \]

      10. +-lowering-+.f64 72.6

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{x + 1}, \frac{y}{\color{blue}{x + 1}}\right)}{t} \]

   8. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, \frac{x}{x + 1}, \frac{y}{x + 1}\right)}{t}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{y}{t}\right) + \frac{y}{t}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{t} + x \cdot \left(1 - \frac{y}{t}\right)} \]

       2. sub-neg N/A

          \[\leadsto \frac{y}{t} + x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{y}{t}\right)\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{y}{t} + x \cdot \left(1 + \color{blue}{-1 \cdot \frac{y}{t}}\right) \]

       4. +-commutative N/A

          \[\leadsto \frac{y}{t} + x \cdot \color{blue}{\left(-1 \cdot \frac{y}{t} + 1\right)} \]

       5. distribute-lft-in N/A

          \[\leadsto \frac{y}{t} + \color{blue}{\left(x \cdot \left(-1 \cdot \frac{y}{t}\right) + x \cdot 1\right)} \]

       6. mul-1-neg N/A

          \[\leadsto \frac{y}{t} + \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{y}{t}\right)\right)} + x \cdot 1\right) \]

       7. distribute-rgt-neg-in N/A

          \[\leadsto \frac{y}{t} + \left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{t}\right)\right)} + x \cdot 1\right) \]

       8. associate-/l* N/A

          \[\leadsto \frac{y}{t} + \left(\left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot y}{t}}\right)\right) + x \cdot 1\right) \]

       9. mul-1-neg N/A

          \[\leadsto \frac{y}{t} + \left(\color{blue}{-1 \cdot \frac{x \cdot y}{t}} + x \cdot 1\right) \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{y}{t} + \left(-1 \cdot \frac{x \cdot y}{t} + \color{blue}{x}\right) \]

       11. associate-+r+ N/A

           \[\leadsto \color{blue}{\left(\frac{y}{t} + -1 \cdot \frac{x \cdot y}{t}\right) + x} \]

   11. Simplified 72.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(z, 0 - \frac{y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 0 - \frac{y}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 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 88.2%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 51.2%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Alternative 21: 2.6% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 -1.0)

C

double code(double x, double y, double z, double t) {
	return -1.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -1.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return -1.0;
}

Python

def code(x, y, z, t):
	return -1.0

Julia

function code(x, y, z, t)
	return -1.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -1.0;
end

Wolfram

code[x_, y_, z_, t_] := -1.0

Derivation

1. Initial program 88.2%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
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 73.5

      \[\leadsto \frac{\frac{y}{t} + x}{\color{blue}{x + 1}} \]

5. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{\frac{y}{t} + x}{x + 1}} \]
6. Step-by-step derivation

   1. unpow1 N/A

      \[\leadsto \frac{\frac{y}{t} + \color{blue}{{x}^{1}}}{x + 1} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{y}{t} + {x}^{\color{blue}{\left(3 - 2\right)}}}{x + 1} \]

   3. pow-div N/A

      \[\leadsto \frac{\frac{y}{t} + \color{blue}{\frac{{x}^{3}}{{x}^{2}}}}{x + 1} \]

   4. sqr-pow N/A

      \[\leadsto \frac{\frac{y}{t} + \frac{\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{{x}^{2}}}{x + 1} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{\frac{y}{t} + \frac{\color{blue}{{\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)}}}{{x}^{2}}}{x + 1} \]

   6. sqr-neg N/A

      \[\leadsto \frac{\frac{y}{t} + \frac{{\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{{x}^{2}}}{x + 1} \]

   7. sub0-neg N/A

      \[\leadsto \frac{\frac{y}{t} + \frac{{\left(\color{blue}{\left(0 - x\right)} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{{x}^{2}}}{x + 1} \]

   8. sub0-neg N/A

      \[\leadsto \frac{\frac{y}{t} + \frac{{\left(\left(0 - x\right) \cdot \color{blue}{\left(0 - x\right)}\right)}^{\left(\frac{3}{2}\right)}}{{x}^{2}}}{x + 1} \]

   9. unpow-prod-down N/A

      \[\leadsto \frac{\frac{y}{t} + \frac{\color{blue}{{\left(0 - x\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x\right)}^{\left(\frac{3}{2}\right)}}}{{x}^{2}}}{x + 1} \]

   10. sqr-pow N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{\color{blue}{{\left(0 - x\right)}^{3}}}{{x}^{2}}}{x + 1} \]

   11. pow2 N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{{\left(0 - x\right)}^{3}}{\color{blue}{x \cdot x}}}{x + 1} \]

   12. sub0-neg N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}}{x \cdot x}}{x + 1} \]

   13. cube-neg N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{x \cdot x}}{x + 1} \]

   14. sub0-neg N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{\color{blue}{0 - {x}^{3}}}{x \cdot x}}{x + 1} \]

   15. +-rgt-identity N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{0 - {x}^{3}}{x \cdot \color{blue}{\left(x + 0\right)}}}{x + 1} \]

   16. distribute-rgt-out N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{0 - {x}^{3}}{\color{blue}{x \cdot x + 0 \cdot x}}}{x + 1} \]

   17. +-lft-identity N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{0 - {x}^{3}}{\color{blue}{0 + \left(x \cdot x + 0 \cdot x\right)}}}{x + 1} \]

   18. metadata-eval N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{0 - {x}^{3}}{\color{blue}{0 \cdot 0} + \left(x \cdot x + 0 \cdot x\right)}}{x + 1} \]

   19. metadata-eval N/A

       \[\leadsto \frac{\frac{y}{t} + \frac{\color{blue}{{0}^{3}} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}}{x + 1} \]

   20. flip3-- N/A

       \[\leadsto \frac{\frac{y}{t} + \color{blue}{\left(0 - x\right)}}{x + 1} \]

   21. sub0-neg N/A

       \[\leadsto \frac{\frac{y}{t} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}{x + 1} \]

   22. sub-neg N/A

       \[\leadsto \frac{\color{blue}{\frac{y}{t} - x}}{x + 1} \]

7. Applied egg-rr 27.0%

   \[\leadsto \frac{\color{blue}{\frac{y}{t} - x}}{x + 1} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1} \]
9. Step-by-step derivation

10. Simplified 2.8%

    \[\leadsto \color{blue}{-1} \]
11. 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 2024198 
(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)))