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

Percentage Accurate: 89.4% → 94.9%

Time: 6.1s

Alternatives: 11

Speedup: 0.5×

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: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+222}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{1 + x} + \frac{y}{\mathsf{fma}\left(t, x, t\right)}\right) - \frac{x}{\mathsf{fma}\left(t, x, t\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 2e+222)
     t_1
     (- (+ (/ x (+ 1.0 x)) (/ y (fma t x t))) (/ x (* (fma t x t) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= 2e+222) {
		tmp = t_1;
	} else {
		tmp = ((x / (1.0 + x)) + (y / fma(t, x, t))) - (x / (fma(t, x, t) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= 2e+222)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(x / Float64(1.0 + x)) + Float64(y / fma(t, x, t))) - Float64(x / Float64(fma(t, x, t) * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+222], t$95$1, N[(N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(t * x + t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e222

   1. Initial program 97.7%

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

   if 2.0000000000000001e222 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 22.3%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{1 + x} + \frac{y}{\mathsf{fma}\left(t, x, t\right)}\right) - \frac{x}{\mathsf{fma}\left(t, x, t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{1 + x}\\ t_2 := t \cdot z - x\\ t_3 := \frac{\frac{z}{1 + x}}{t\_2} \cdot y\\ t_4 := \frac{\frac{z \cdot y - x}{t\_2} + x}{1 + x}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (- (* t z) x))
        (t_3 (* (/ (/ z (+ 1.0 x)) t_2) y))
        (t_4 (/ (+ (/ (- (* z y) x) t_2) x) (+ 1.0 x))))
   (if (<= t_4 -2e+18)
     t_3
     (if (<= t_4 5e-73)
       t_1
       (if (<= t_4 2.0)
         (/ (- x (/ x (fma t z (- x)))) (+ 1.0 x))
         (if (<= t_4 INFINITY) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (t * z) - x;
	double t_3 = ((z / (1.0 + x)) / t_2) * y;
	double t_4 = ((((z * y) - x) / t_2) + x) / (1.0 + x);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 5e-73) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / fma(t, z, -x))) / (1.0 + x);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(Float64(z / Float64(1.0 + x)) / t_2) * y)
	t_4 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / t_2) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 5e-73)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(1.0 + x));
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+18], t$95$3, If[LessEqual[t$95$4, 5e-73], t$95$1, If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, 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))) < -2e18 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 82.0%

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

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 96.5%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999998e-73 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if 4.9999999999999998e-73 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   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

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{z}{1 + x}}{t \cdot z - x} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{1 + x}}{t \cdot z - x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{1 + x}\\ t_2 := t \cdot z - x\\ t_3 := \frac{z}{\left(1 + x\right) \cdot t\_2} \cdot y\\ t_4 := \frac{\frac{z \cdot y - x}{t\_2} + x}{1 + x}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (- (* t z) x))
        (t_3 (* (/ z (* (+ 1.0 x) t_2)) y))
        (t_4 (/ (+ (/ (- (* z y) x) t_2) x) (+ 1.0 x))))
   (if (<= t_4 -2e+18)
     t_3
     (if (<= t_4 5e-73)
       t_1
       (if (<= t_4 2.0)
         (/ (- x (/ x (fma t z (- x)))) (+ 1.0 x))
         (if (<= t_4 INFINITY) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (t * z) - x;
	double t_3 = (z / ((1.0 + x) * t_2)) * y;
	double t_4 = ((((z * y) - x) / t_2) + x) / (1.0 + x);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 5e-73) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / fma(t, z, -x))) / (1.0 + x);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(z / Float64(Float64(1.0 + x) * t_2)) * y)
	t_4 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / t_2) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 5e-73)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(1.0 + x));
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+18], t$95$3, If[LessEqual[t$95$4, 5e-73], t$95$1, If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, 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))) < -2e18 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 82.0%

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

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 96.4%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999998e-73 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if 4.9999999999999998e-73 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   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

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{1 + x}\\ t_2 := t \cdot z - x\\ t_3 := \frac{z}{\left(1 + x\right) \cdot t\_2} \cdot y\\ t_4 := \frac{\frac{z \cdot y - x}{t\_2} + x}{1 + x}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (- (* t z) x))
        (t_3 (* (/ z (* (+ 1.0 x) t_2)) y))
        (t_4 (/ (+ (/ (- (* z y) x) t_2) x) (+ 1.0 x))))
   (if (<= t_4 -2e+18)
     t_3
     (if (<= t_4 2e-11)
       t_1
       (if (<= t_4 2.0) 1.0 (if (<= t_4 INFINITY) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (t * z) - x;
	double t_3 = (z / ((1.0 + x) * t_2)) * y;
	double t_4 = ((((z * y) - x) / t_2) + x) / (1.0 + x);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 2e-11) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = 1.0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = (t * z) - x;
	double t_3 = (z / ((1.0 + x) * t_2)) * y;
	double t_4 = ((((z * y) - x) / t_2) + x) / (1.0 + x);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 2e-11) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = 1.0;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (1.0 + x)
	t_2 = (t * z) - x
	t_3 = (z / ((1.0 + x) * t_2)) * y
	t_4 = ((((z * y) - x) / t_2) + x) / (1.0 + x)
	tmp = 0
	if t_4 <= -2e+18:
		tmp = t_3
	elif t_4 <= 2e-11:
		tmp = t_1
	elif t_4 <= 2.0:
		tmp = 1.0
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(z / Float64(Float64(1.0 + x) * t_2)) * y)
	t_4 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / t_2) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 2e-11)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = 1.0;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (1.0 + x);
	t_2 = (t * z) - x;
	t_3 = (z / ((1.0 + x) * t_2)) * y;
	t_4 = ((((z * y) - x) / t_2) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_4 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 2e-11)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = 1.0;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+18], t$95$3, If[LessEqual[t$95$4, 2e-11], t$95$1, If[LessEqual[t$95$4, 2.0], 1.0, If[LessEqual[t$95$4, Infinity], t$95$3, 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))) < -2e18 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 82.0%

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

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 96.4%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999988e-11 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 1.99999999999999988e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq \infty:\\ \;\;\;\;\frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(1 + x\right) \cdot t}\\ t_2 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (+ 1.0 x) t)))
        (t_2 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_2 -2e-13)
     t_1
     (if (<= t_2 2e-11) (* (- 1.0 x) x) (if (<= t_2 2.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_2 <= -2e-13) {
		tmp = t_1;
	} else if (t_2 <= 2e-11) {
		tmp = (1.0 - x) * x;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y / ((1.0d0 + x) * t)
    t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_2 <= (-2d-13)) then
        tmp = t_1
    else if (t_2 <= 2d-11) then
        tmp = (1.0d0 - x) * x
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_2 <= -2e-13) {
		tmp = t_1;
	} else if (t_2 <= 2e-11) {
		tmp = (1.0 - x) * x;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / ((1.0 + x) * t)
	t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_2 <= -2e-13:
		tmp = t_1
	elif t_2 <= 2e-11:
		tmp = (1.0 - x) * x
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(1.0 + x) * t))
	t_2 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= -2e-13)
		tmp = t_1;
	elseif (t_2 <= 2e-11)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / ((1.0 + x) * t);
	t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= -2e-13)
		tmp = t_1;
	elseif (t_2 <= 2e-11)
		tmp = (1.0 - x) * x;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-13], t$95$1, If[LessEqual[t$95$2, 2e-11], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.0000000000000001e-13 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.3%

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

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 64.0%

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

   if -2.0000000000000001e-13 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999988e-11

   1. Initial program 97.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. lower-+.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.1%

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

   if 1.99999999999999988e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 -2e-13)
     (/ y t)
     (if (<= t_1 2e-11) (* (- 1.0 x) x) (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e-13) {
		tmp = y / t;
	} else if (t_1 <= 2e-11) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_1 <= (-2d-13)) then
        tmp = y / t
    else if (t_1 <= 2d-11) then
        tmp = (1.0d0 - x) * x
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= -2e-13) {
		tmp = y / t;
	} else if (t_1 <= 2e-11) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_1 <= -2e-13:
		tmp = y / t
	elif t_1 <= 2e-11:
		tmp = (1.0 - x) * x
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -2e-13)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-11)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -2e-13)
		tmp = y / t;
	elseif (t_1 <= 2e-11)
		tmp = (1.0 - x) * x;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-13], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.0000000000000001e-13 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 52.3

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

   5. Applied rewrites 52.3%

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

   if -2.0000000000000001e-13 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999988e-11

   1. Initial program 97.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. lower-+.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.1%

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

   if 1.99999999999999988e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{1 + x}\\ t_2 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_2 2e-11) t_1 (if (<= t_2 1.0) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_2 <= 2e-11) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((y / t) + x) / (1.0d0 + x)
    t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_2 <= 2d-11) then
        tmp = t_1
    else if (t_2 <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (1.0 + x);
	double t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_2 <= 2e-11) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (1.0 + x)
	t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_2 <= 2e-11:
		tmp = t_1
	elif t_2 <= 1.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x))
	t_2 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= 2e-11)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (1.0 + x);
	t_2 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= 2e-11)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-11], t$95$1, If[LessEqual[t$95$2, 1.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999988e-11 or 1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if 1.99999999999999988e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 2e-11)
     (/ (+ (/ y t) x) 1.0)
     (if (<= t_1 2.0) 1.0 (/ y (* (+ 1.0 x) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= 2e-11) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / ((1.0 + x) * 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 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_1 <= 2d-11) then
        tmp = ((y / t) + x) / 1.0d0
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / ((1.0d0 + x) * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= 2e-11) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / ((1.0 + x) * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_1 <= 2e-11:
		tmp = ((y / t) + x) / 1.0
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / ((1.0 + x) * t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= 2e-11)
		tmp = Float64(Float64(Float64(y / t) + x) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / Float64(Float64(1.0 + x) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= 2e-11)
		tmp = ((y / t) + x) / 1.0;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / ((1.0 + x) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-11], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999988e-11

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.3%

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

   if 1.99999999999999988e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

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

   1. Initial program 56.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 60.9

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 59.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+222}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x))))
   (if (<= t_1 2e+222) t_1 (/ (+ (/ y t) x) (+ 1.0 x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= 2e+222) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	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 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0d0 + x)
    if (t_1 <= 2d+222) then
        tmp = t_1
    else
        tmp = ((y / t) + x) / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	double tmp;
	if (t_1 <= 2e+222) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)
	tmp = 0
	if t_1 <= 2e+222:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= 2e+222)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= 2e+222)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+222], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e222

   1. Initial program 97.7%

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

   if 2.0000000000000001e222 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 22.3%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 91.6

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

   5. Applied rewrites 91.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ (/ (- (* z y) x) (- (* t z) x)) x) (+ 1.0 x)) 2e-11)
   (* (- 1.0 x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 2e-11) {
		tmp = (1.0 - 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 ((((((z * y) - x) / ((t * z) - x)) + x) / (1.0d0 + x)) <= 2d-11) then
        tmp = (1.0d0 - 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 ((((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 2e-11) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 2e-11:
		tmp = (1.0 - x) * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(z * y) - x) / Float64(Float64(t * z) - x)) + x) / Float64(1.0 + x)) <= 2e-11)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((((((z * y) - x) / ((t * z) - x)) + x) / (1.0 + x)) <= 2e-11)
		tmp = (1.0 - x) * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 2e-11], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999988e-11

   1. Initial program 94.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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. lower-+.f64 37.6

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

   5. Applied rewrites 37.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.1%

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

   if 1.99999999999999988e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 84.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 69.7%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y - x}{t \cdot z - x} + x}{1 + x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.0% 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 87.7%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 47.7%

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

Developer Target 1: 99.6% 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 2024332 
(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)))