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

Percentage Accurate: 89.5% → 97.3%

Time: 11.1s

Alternatives: 14

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_2 := x - z \cdot t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+139}:\\ \;\;\;\;\frac{x + y \cdot \left(\frac{x}{y \cdot t\_2} - \frac{z}{t\_2}\right)}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_2 (- x (* z t))))
   (if (<= t_1 -4e+139)
     (/ (+ x (* y (- (/ x (* y t_2)) (/ z t_2)))) (+ x 1.0))
     (if (<= t_1 2e+263) t_1 (+ (/ x (+ x 1.0)) (/ y (fma x t t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_2 = x - (z * t);
	double tmp;
	if (t_1 <= -4e+139) {
		tmp = (x + (y * ((x / (y * t_2)) - (z / t_2)))) / (x + 1.0);
	} else if (t_1 <= 2e+263) {
		tmp = t_1;
	} else {
		tmp = (x / (x + 1.0)) + (y / fma(x, t, t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_2 = Float64(x - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -4e+139)
		tmp = Float64(Float64(x + Float64(y * Float64(Float64(x / Float64(y * t_2)) - Float64(z / t_2)))) / Float64(x + 1.0));
	elseif (t_1 <= 2e+263)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / fma(x, t, t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+139], N[(N[(x + N[(y * N[(N[(x / N[(y * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+263], t$95$1, N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000013e139

   1. Initial program 63.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   if -4.00000000000000013e139 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000003e263

   1. Initial program 98.9%

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

   if 2.00000000000000003e263 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 28.5%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 88.1

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

   5. Simplified 88.1%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 88.1

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

   8. Simplified 88.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -4 \cdot 10^{+139}:\\ \;\;\;\;\frac{x + y \cdot \left(\frac{x}{y \cdot \left(x - z \cdot t\right)} - \frac{z}{x - z \cdot t}\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_3 := x - z \cdot t\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.2:\\ \;\;\;\;\frac{x + \frac{x}{t\_3}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{y \cdot z}{t\_3 \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x (+ x 1.0)) (/ y (fma x t t))))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_3 (- x (* z t))))
   (if (<= t_2 2e-61)
     t_1
     (if (<= t_2 1.2)
       (/ (+ x (/ x t_3)) (+ x 1.0))
       (if (<= t_2 2e+263) (/ (* y z) (* t_3 (- -1.0 x))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / (x + 1.0)) + (y / fma(x, t, t));
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_3 = x - (z * t);
	double tmp;
	if (t_2 <= 2e-61) {
		tmp = t_1;
	} else if (t_2 <= 1.2) {
		tmp = (x + (x / t_3)) / (x + 1.0);
	} else if (t_2 <= 2e+263) {
		tmp = (y * z) / (t_3 * (-1.0 - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / fma(x, t, t)))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_3 = Float64(x - Float64(z * t))
	tmp = 0.0
	if (t_2 <= 2e-61)
		tmp = t_1;
	elseif (t_2 <= 1.2)
		tmp = Float64(Float64(x + Float64(x / t_3)) / Float64(x + 1.0));
	elseif (t_2 <= 2e+263)
		tmp = Float64(Float64(y * z) / Float64(t_3 * Float64(-1.0 - x)));
	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[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-61], t$95$1, If[LessEqual[t$95$2, 1.2], N[(N[(x + N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+263], N[(N[(y * z), $MachinePrecision] / N[(t$95$3 * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-61 or 2.00000000000000003e263 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.2%

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

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 88.2

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

   5. Simplified 88.2%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 81.5

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

   8. Simplified 81.5%

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

   if 2.0000000000000001e-61 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.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. lower-+.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 1.19999999999999996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000003e263

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{y \cdot z}{\left(x - z \cdot t\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 0.98:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{y \cdot z}{\left(x - z \cdot t\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x (+ x 1.0)) (/ y (fma x t t))))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 0.98)
     t_1
     (if (<= t_2 1.2)
       1.0
       (if (<= t_2 2e+263) (/ (* y z) (* (- x (* z t)) (- -1.0 x))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / (x + 1.0)) + (y / fma(x, t, t));
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.98) {
		tmp = t_1;
	} else if (t_2 <= 1.2) {
		tmp = 1.0;
	} else if (t_2 <= 2e+263) {
		tmp = (y * z) / ((x - (z * t)) * (-1.0 - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / fma(x, t, t)))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 0.98)
		tmp = t_1;
	elseif (t_2 <= 1.2)
		tmp = 1.0;
	elseif (t_2 <= 2e+263)
		tmp = Float64(Float64(y * z) / Float64(Float64(x - Float64(z * t)) * Float64(-1.0 - x)));
	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[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.98], t$95$1, If[LessEqual[t$95$2, 1.2], 1.0, If[LessEqual[t$95$2, 2e+263], N[(N[(y * z), $MachinePrecision] / N[(N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < 0.97999999999999998 or 2.00000000000000003e263 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.3%

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

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 88.9

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

   5. Simplified 88.9%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 81.2

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

   8. Simplified 81.2%

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

   if 0.97999999999999998 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.4%

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

   if 1.19999999999999996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000003e263

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.98:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{y \cdot z}{\left(x - z \cdot t\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 0.98:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{y \cdot z}{\left(x - z \cdot t\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 0.98)
     t_1
     (if (<= t_2 1.2)
       1.0
       (if (<= t_2 2e+263) (/ (* y z) (* (- x (* z t)) (- -1.0 x))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.98) {
		tmp = t_1;
	} else if (t_2 <= 1.2) {
		tmp = 1.0;
	} else if (t_2 <= 2e+263) {
		tmp = (y * z) / ((x - (z * t)) * (-1.0 - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_2 <= 0.98d0) then
        tmp = t_1
    else if (t_2 <= 1.2d0) then
        tmp = 1.0d0
    else if (t_2 <= 2d+263) then
        tmp = (y * z) / ((x - (z * t)) * ((-1.0d0) - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.98) {
		tmp = t_1;
	} else if (t_2 <= 1.2) {
		tmp = 1.0;
	} else if (t_2 <= 2e+263) {
		tmp = (y * z) / ((x - (z * t)) * (-1.0 - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= 0.98:
		tmp = t_1
	elif t_2 <= 1.2:
		tmp = 1.0
	elif t_2 <= 2e+263:
		tmp = (y * z) / ((x - (z * t)) * (-1.0 - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 0.98)
		tmp = t_1;
	elseif (t_2 <= 1.2)
		tmp = 1.0;
	elseif (t_2 <= 2e+263)
		tmp = Float64(Float64(y * z) / Float64(Float64(x - Float64(z * t)) * Float64(-1.0 - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 0.98)
		tmp = t_1;
	elseif (t_2 <= 1.2)
		tmp = 1.0;
	elseif (t_2 <= 2e+263)
		tmp = (y * z) / ((x - (z * t)) * (-1.0 - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.98], t$95$1, If[LessEqual[t$95$2, 1.2], 1.0, If[LessEqual[t$95$2, 2e+263], N[(N[(y * z), $MachinePrecision] / N[(N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < 0.97999999999999998 or 2.00000000000000003e263 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 80.4

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

   5. Simplified 80.4%

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

   if 0.97999999999999998 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.4%

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

   if 1.19999999999999996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000003e263

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.98:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{y \cdot z}{\left(x - z \cdot t\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+41}:\\ \;\;\;\;t\_1 + 1\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;t\_1 + \mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;t\_2 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -2e+41)
     (+ t_1 1.0)
     (if (<= t_2 0.02) (+ t_1 (fma x (- x) x)) (if (<= t_2 1.2) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+41) {
		tmp = t_1 + 1.0;
	} else if (t_2 <= 0.02) {
		tmp = t_1 + fma(x, -x, x);
	} else if (t_2 <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e+41)
		tmp = Float64(t_1 + 1.0);
	elseif (t_2 <= 0.02)
		tmp = Float64(t_1 + fma(x, Float64(-x), x));
	elseif (t_2 <= 1.2)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+41], N[(t$95$1 + 1.0), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(t$95$1 + N[(x * (-x) + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.2], 1.0, t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000001e41

   1. Initial program 75.7%

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

   3. Taylor expanded in 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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 75.8

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

   8. Simplified 75.8%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 75.8%

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

   if -2.00000000000000001e41 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 80.0

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

   8. Simplified 80.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 78.3

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

   11. Simplified 78.3%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.8%

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

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

   1. Initial program 53.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 77.6

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

   8. Simplified 77.6%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 64.3

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

   11. Simplified 64.3%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)} + 1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.02:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)} + \mathsf{fma}\left(x, -x, x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+41}:\\ \;\;\;\;t\_1 + 1\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, 1 - x, x\right)\\ \mathbf{elif}\;t\_2 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -2e+41)
     (+ t_1 1.0)
     (if (<= t_2 0.02) (fma (/ y t) (- 1.0 x) x) (if (<= t_2 1.2) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+41) {
		tmp = t_1 + 1.0;
	} else if (t_2 <= 0.02) {
		tmp = fma((y / t), (1.0 - x), x);
	} else if (t_2 <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e+41)
		tmp = Float64(t_1 + 1.0);
	elseif (t_2 <= 0.02)
		tmp = fma(Float64(y / t), Float64(1.0 - x), x);
	elseif (t_2 <= 1.2)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+41], N[(t$95$1 + 1.0), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(y / t), $MachinePrecision] * N[(1.0 - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 1.2], 1.0, t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000001e41

   1. Initial program 75.7%

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

   3. Taylor expanded in 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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 75.8

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

   8. Simplified 75.8%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 75.8%

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

   if -2.00000000000000001e41 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 80.0

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

   8. Simplified 80.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.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. lower-fma.f64 N/A

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

       5. lower-/.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. lower-+.f64 N/A

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

       8. lower-/.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. lower-+.f64 79.9

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

   11. Simplified 79.9%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{x \cdot \left(1 - \frac{y}{t}\right) + \frac{y}{t}} \]
   13. 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

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

       12. mul-1-neg N/A

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

       13. associate-/l* N/A

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

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

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

       15. *-lft-identity N/A

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

       16. mul-1-neg N/A

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

       17. distribute-rgt-out N/A

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

       18. lower-fma.f64 N/A

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

   14. Simplified 76.7%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.8%

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

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

   1. Initial program 53.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 77.6

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

   8. Simplified 77.6%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 64.3

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

   11. Simplified 64.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)} + 1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, 1 - x, x\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;t\_1 + 1\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -1e+19)
     (+ t_1 1.0)
     (if (<= t_2 0.98) (/ x (+ x 1.0)) (if (<= t_2 1.2) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -1e+19) {
		tmp = t_1 + 1.0;
	} else if (t_2 <= 0.98) {
		tmp = x / (x + 1.0);
	} else if (t_2 <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1e+19)
		tmp = Float64(t_1 + 1.0);
	elseif (t_2 <= 0.98)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_2 <= 1.2)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+19], N[(t$95$1 + 1.0), $MachinePrecision], If[LessEqual[t$95$2, 0.98], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.2], 1.0, t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e19

   1. Initial program 78.3%

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

   3. Taylor expanded in 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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 75.8

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

   8. Simplified 75.8%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 75.8%

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

   if -1e19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.97999999999999998

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 53.2

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

   5. Simplified 53.2%

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

   if 0.97999999999999998 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.4%

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

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

   1. Initial program 53.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 77.6

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

   8. Simplified 77.6%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 64.3

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

   11. Simplified 64.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)} + 1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.98:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma x t t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -5e-12)
     t_1
     (if (<= t_2 0.98) (/ x (+ x 1.0)) (if (<= t_2 1.2) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(x, t, t);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e-12) {
		tmp = t_1;
	} else if (t_2 <= 0.98) {
		tmp = x / (x + 1.0);
	} else if (t_2 <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(x, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5e-12)
		tmp = t_1;
	elseif (t_2 <= 0.98)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_2 <= 1.2)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-12], t$95$1, If[LessEqual[t$95$2, 0.98], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.2], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999997e-12 or 1.19999999999999996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 65.4%

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

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 76.7

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

   5. Simplified 76.7%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 76.8

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

   8. Simplified 76.8%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 64.6

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

   11. Simplified 64.6%

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

   if -4.9999999999999997e-12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.97999999999999998

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 55.3

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

   5. Simplified 55.3%

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

   if 0.97999999999999998 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.98:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.98:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -5e-12)
     (/ y t)
     (if (<= t_1 0.98) (/ x (+ x 1.0)) (if (<= t_1 1.2) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e-12) {
		tmp = y / t;
	} else if (t_1 <= 0.98) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= (-5d-12)) then
        tmp = y / t
    else if (t_1 <= 0.98d0) then
        tmp = x / (x + 1.0d0)
    else if (t_1 <= 1.2d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e-12) {
		tmp = y / t;
	} else if (t_1 <= 0.98) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -5e-12:
		tmp = y / t
	elif t_1 <= 0.98:
		tmp = x / (x + 1.0)
	elif t_1 <= 1.2:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -5e-12)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.98)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_1 <= 1.2)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -5e-12)
		tmp = y / t;
	elseif (t_1 <= 0.98)
		tmp = x / (x + 1.0);
	elseif (t_1 <= 1.2)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-12], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.98], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.2], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999997e-12 or 1.19999999999999996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 65.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 58.2

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

   5. Simplified 58.2%

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

   if -4.9999999999999997e-12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.97999999999999998

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 55.3

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

   5. Simplified 55.3%

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

   if 0.97999999999999998 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.98:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_1 \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_1 -5e-12)
     (/ y t)
     (if (<= t_1 0.02) (- x (* x x)) (if (<= t_1 1.2) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e-12) {
		tmp = y / t;
	} else if (t_1 <= 0.02) {
		tmp = x - (x * x);
	} else if (t_1 <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_1 <= (-5d-12)) then
        tmp = y / t
    else if (t_1 <= 0.02d0) then
        tmp = x - (x * x)
    else if (t_1 <= 1.2d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e-12) {
		tmp = y / t;
	} else if (t_1 <= 0.02) {
		tmp = x - (x * x);
	} else if (t_1 <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -5e-12:
		tmp = y / t
	elif t_1 <= 0.02:
		tmp = x - (x * x)
	elif t_1 <= 1.2:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -5e-12)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.02)
		tmp = Float64(x - Float64(x * x));
	elseif (t_1 <= 1.2)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -5e-12)
		tmp = y / t;
	elseif (t_1 <= 0.02)
		tmp = x - (x * x);
	elseif (t_1 <= 1.2)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-12], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.2], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999997e-12 or 1.19999999999999996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 65.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 58.2

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

   5. Simplified 58.2%

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

   if -4.9999999999999997e-12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 54.4

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

   5. Simplified 54.4%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

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

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 53.3

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

   8. Simplified 53.3%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.02:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_2 := \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 + 1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_2 (/ y (fma x t t))))
   (if (<= t_1 (- INFINITY))
     (+ t_2 1.0)
     (if (<= t_1 2e+263) t_1 (+ (/ x (+ x 1.0)) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_2 = y / fma(x, t, t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 + 1.0;
	} else if (t_1 <= 2e+263) {
		tmp = t_1;
	} else {
		tmp = (x / (x + 1.0)) + t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_2 = Float64(y / fma(x, t, t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_2 + 1.0);
	elseif (t_1 <= 2e+263)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / N[(x * t + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 + 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+263], t$95$1, N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 45.0%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 85.8

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

   5. Simplified 85.8%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 85.8

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

   8. Simplified 85.8%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 85.8%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000003e263

   1. Initial program 99.0%

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

   if 2.00000000000000003e263 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 28.5%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 88.1

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

   5. Simplified 88.1%

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

   6. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-fma.f64 88.1

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

   8. Simplified 88.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, t, t\right)} + 1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.02:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)) 0.02)
   (- x (* x x))
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 0.02) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)) <= 0.02d0) then
        tmp = x - (x * x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 0.02) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 0.02:
		tmp = x - (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0)) <= 0.02)
		tmp = Float64(x - Float64(x * x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 0.02)
		tmp = x - (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 0.02], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 89.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 36.7

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

   5. Simplified 36.7%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

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

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

      5. unpow2 N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 31.7

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

   8. Simplified 31.7%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 85.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.4%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 0.02:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{-128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.37:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z}{\mathsf{fma}\left(x, x, x\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t -1.35e-128)
     t_1
     (if (<= t 0.37) (fma (- y) (/ z (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 <= -1.35e-128) {
		tmp = t_1;
	} else if (t <= 0.37) {
		tmp = fma(-y, (z / 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 <= -1.35e-128)
		tmp = t_1;
	elseif (t <= 0.37)
		tmp = fma(Float64(-y), Float64(z / 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, -1.35e-128], t$95$1, If[LessEqual[t, 0.37], N[((-y) * N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35000000000000003e-128 or 0.37 < t

   1. Initial program 82.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 91.1

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

   5. Simplified 91.1%

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

   if -1.35000000000000003e-128 < t < 0.37

   1. Initial program 94.0%

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

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in z 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. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 81.3

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

   8. Simplified 81.3%

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

Alternative 14: 54.2% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 49.9%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024215 
(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)))