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

Percentage Accurate: 88.8% → 96.5%

Time: 6.4s

Alternatives: 13

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: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{\frac{z}{t\_2} \cdot y}{1 + x}\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x)))
        (t_2 (fma t z (- x)))
        (t_3 (/ (* (/ z t_2) y) (+ 1.0 x))))
   (if (<= t_1 -200.0)
     t_3
     (if (<= t_1 1e-13)
       (/ (- x (/ (- (/ x z) y) t)) (+ 1.0 x))
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ 1.0 x))
         (if (<= t_1 INFINITY) t_3 (/ (+ (/ y t) x) (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double t_2 = fma(t, z, -x);
	double t_3 = ((z / t_2) * y) / (1.0 + x);
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_3;
	} else if (t_1 <= 1e-13) {
		tmp = (x - (((x / z) - y) / t)) / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(Float64(z / t_2) * y) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_3;
	elseif (t_1 <= 1e-13)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(1.0 + x));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(1.0 + x));
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$3, If[LessEqual[t$95$1, 1e-13], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -200 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 75.1%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -200 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-13

   1. Initial program 90.7%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. 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 99.6

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq -200:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 10^{-13}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{\frac{z}{t\_2} \cdot y}{1 + x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\left(1 - x\right) \cdot x + \frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x)))
        (t_2 (fma t z (- x)))
        (t_3 (/ (* (/ z t_2) y) (+ 1.0 x))))
   (if (<= t_1 -5e-14)
     t_3
     (if (<= t_1 1e-13)
       (+ (* (- 1.0 x) x) (/ y (* (+ 1.0 x) t)))
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ 1.0 x))
         (if (<= t_1 INFINITY) t_3 (/ (+ (/ y t) x) (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double t_2 = fma(t, z, -x);
	double t_3 = ((z / t_2) * y) / (1.0 + x);
	double tmp;
	if (t_1 <= -5e-14) {
		tmp = t_3;
	} else if (t_1 <= 1e-13) {
		tmp = ((1.0 - x) * x) + (y / ((1.0 + x) * t));
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = ((y / t) + x) / (1.0 + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(Float64(z / t_2) * y) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e-14)
		tmp = t_3;
	elseif (t_1 <= 1e-13)
		tmp = Float64(Float64(Float64(1.0 - x) * x) + Float64(y / Float64(Float64(1.0 + x) * t)));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(1.0 + x));
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z / t$95$2), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-14], t$95$3, If[LessEqual[t$95$1, 1e-13], N[(N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision] + N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.0000000000000002e-14 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 75.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if -5.0000000000000002e-14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-13

   1. Initial program 90.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. 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 99.6

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 10^{-13}:\\ \;\;\;\;\left(1 - x\right) \cdot x + \frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq \infty:\\ \;\;\;\;\frac{\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{t\_2}\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\left(1 - x\right) \cdot x + t\_3\\ \mathbf{elif}\;t\_1 \leq 2000000000000:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x)))
        (t_2 (fma t z (- x)))
        (t_3 (/ y (* (+ 1.0 x) t))))
   (if (<= t_1 -5e-14)
     (* (/ z (+ 1.0 x)) (/ y t_2))
     (if (<= t_1 1e-13)
       (+ (* (- 1.0 x) x) t_3)
       (if (<= t_1 2000000000000.0)
         (/ (- x (/ x t_2)) (+ 1.0 x))
         (+ t_3 (/ x (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double t_2 = fma(t, z, -x);
	double t_3 = y / ((1.0 + x) * t);
	double tmp;
	if (t_1 <= -5e-14) {
		tmp = (z / (1.0 + x)) * (y / t_2);
	} else if (t_1 <= 1e-13) {
		tmp = ((1.0 - x) * x) + t_3;
	} else if (t_1 <= 2000000000000.0) {
		tmp = (x - (x / t_2)) / (1.0 + x);
	} else {
		tmp = t_3 + (x / (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(y / Float64(Float64(1.0 + x) * t))
	tmp = 0.0
	if (t_1 <= -5e-14)
		tmp = Float64(Float64(z / Float64(1.0 + x)) * Float64(y / t_2));
	elseif (t_1 <= 1e-13)
		tmp = Float64(Float64(Float64(1.0 - x) * x) + t_3);
	elseif (t_1 <= 2000000000000.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(1.0 + x));
	else
		tmp = Float64(t_3 + Float64(x / Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-14], N[(N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-13], N[(N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$1, 2000000000000.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.0000000000000002e-14

   1. Initial program 83.6%

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

   3. Taylor expanded in 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 74.9

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

   5. Applied rewrites 74.9%

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

   if -5.0000000000000002e-14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-13

   1. Initial program 90.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.8%

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

   if 1e-13 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. 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 98.9

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

   5. Applied rewrites 98.9%

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

   if 2e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 53.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 70.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{\mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 10^{-13}:\\ \;\;\;\;\left(1 - x\right) \cdot x + \frac{y}{\left(1 + x\right) \cdot t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 2000000000000:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t} + \frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(1 + x\right) \cdot t} + \frac{x}{1 + x}\\ t_2 := \frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x}\\ \mathbf{if}\;t\_2 \leq 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2000000000000:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y (* (+ 1.0 x) t)) (/ x (+ 1.0 x))))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x))))
   (if (<= t_2 1e-13)
     t_1
     (if (<= t_2 2000000000000.0)
       (/ (- x (/ x (fma t z (- x)))) (+ 1.0 x))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / ((1.0 + x) * t)) + (x / (1.0 + x));
	double t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 1e-13) {
		tmp = t_1;
	} else if (t_2 <= 2000000000000.0) {
		tmp = (x - (x / fma(t, z, -x))) / (1.0 + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / Float64(Float64(1.0 + x) * t)) + Float64(x / Float64(1.0 + x)))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= 1e-13)
		tmp = t_1;
	elseif (t_2 <= 2000000000000.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(1.0 + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-13], t$95$1, If[LessEqual[t$95$2, 2000000000000.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 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))) < 1e-13 or 2e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.7%

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

   if 1e-13 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. 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 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 10^{-13}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t} + \frac{x}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 2000000000000:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t} + \frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(1 + x\right) \cdot t} + \frac{x}{1 + x}\\ t_2 := \frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x}\\ \mathbf{if}\;t\_2 \leq 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2000000000000:\\ \;\;\;\;\frac{\left(x - \frac{z \cdot y}{x}\right) + 1}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y (* (+ 1.0 x) t)) (/ x (+ 1.0 x))))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x))))
   (if (<= t_2 1e-13)
     t_1
     (if (<= t_2 2000000000000.0)
       (/ (+ (- x (/ (* z y) x)) 1.0) (+ 1.0 x))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / ((1.0 + x) * t)) + (x / (1.0 + x));
	double t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 1e-13) {
		tmp = t_1;
	} else if (t_2 <= 2000000000000.0) {
		tmp = ((x - ((z * y) / x)) + 1.0) / (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 = (y / ((1.0d0 + x) * t)) + (x / (1.0d0 + x))
    t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0d0 + x)
    if (t_2 <= 1d-13) then
        tmp = t_1
    else if (t_2 <= 2000000000000.0d0) then
        tmp = ((x - ((z * y) / x)) + 1.0d0) / (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 = (y / ((1.0 + x) * t)) + (x / (1.0 + x));
	double t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 1e-13) {
		tmp = t_1;
	} else if (t_2 <= 2000000000000.0) {
		tmp = ((x - ((z * y) / x)) + 1.0) / (1.0 + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / ((1.0 + x) * t)) + (x / (1.0 + x))
	t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x)
	tmp = 0
	if t_2 <= 1e-13:
		tmp = t_1
	elif t_2 <= 2000000000000.0:
		tmp = ((x - ((z * y) / x)) + 1.0) / (1.0 + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / Float64(Float64(1.0 + x) * t)) + Float64(x / Float64(1.0 + x)))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= 1e-13)
		tmp = t_1;
	elseif (t_2 <= 2000000000000.0)
		tmp = Float64(Float64(Float64(x - Float64(Float64(z * y) / x)) + 1.0) / Float64(1.0 + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / ((1.0 + x) * t)) + (x / (1.0 + x));
	t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= 1e-13)
		tmp = t_1;
	elseif (t_2 <= 2000000000000.0)
		tmp = ((x - ((z * y) / x)) + 1.0) / (1.0 + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-13], t$95$1, If[LessEqual[t$95$2, 2000000000000.0], N[(N[(N[(x - N[(N[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 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))) < 1e-13 or 2e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.7%

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

   if 1e-13 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e12

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 10^{-13}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t} + \frac{x}{1 + x}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 2000000000000:\\ \;\;\;\;\frac{\left(x - \frac{z \cdot y}{x}\right) + 1}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t} + \frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y (* (+ 1.0 x) t)) (/ x (+ 1.0 x))))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x))))
   (if (<= t_2 1e-13) 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 / ((1.0 + x) * t)) + (x / (1.0 + x));
	double t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 1e-13) {
		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 / ((1.0d0 + x) * t)) + (x / (1.0d0 + x))
    t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0d0 + x)
    if (t_2 <= 1d-13) 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 / ((1.0 + x) * t)) + (x / (1.0 + x));
	double t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 1e-13) {
		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 / ((1.0 + x) * t)) + (x / (1.0 + x))
	t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x)
	tmp = 0
	if t_2 <= 1e-13:
		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(y / Float64(Float64(1.0 + x) * t)) + Float64(x / Float64(1.0 + x)))
	t_2 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= 1e-13)
		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 / ((1.0 + x) * t)) + (x / (1.0 + x));
	t_2 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= 1e-13)
		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[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-13], 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))) < 1e-13 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 75.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.4%

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

   if 1e-13 < (/.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 98.8%

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

4. Final simplification 85.4%

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

Alternative 7: 85.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ 1.0 x)))
        (t_2 (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x))))
   (if (<= t_2 1e-13) 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 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 1e-13) {
		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 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0d0 + x)
    if (t_2 <= 1d-13) 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 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_2 <= 1e-13) {
		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 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x)
	tmp = 0
	if t_2 <= 1e-13:
		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(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= 1e-13)
		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 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= 1e-13)
		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[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-13], 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))) < 1e-13 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 75.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 73.3

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

   5. Applied rewrites 73.3%

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

   if 1e-13 < (/.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 98.8%

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

4. Final simplification 85.4%

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

Alternative 8: 71.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x}\\ \mathbf{if}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x))))
   (if (<= t_1 1e-13) (/ y t) (if (<= t_1 2e+16) 1.0 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= 1e-13) {
		tmp = y / t;
	} else if (t_1 <= 2e+16) {
		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 - ((x - (z * y)) / ((z * t) - x))) / (1.0d0 + x)
    if (t_1 <= 1d-13) then
        tmp = y / t
    else if (t_1 <= 2d+16) 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 - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= 1e-13) {
		tmp = y / t;
	} else if (t_1 <= 2e+16) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x)
	tmp = 0
	if t_1 <= 1e-13:
		tmp = y / t
	elif t_1 <= 2e+16:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= 1e-13)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e+16)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= 1e-13)
		tmp = y / t;
	elseif (t_1 <= 2e+16)
		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[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-13], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e+16], 1.0, N[(y / t), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-13 or 2e16 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 46.1

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

   5. Applied rewrites 46.1%

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

   if 1e-13 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 10^{-13}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, -x\right)\\ t_2 := \frac{x}{1 + x}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t\_1}, \frac{z}{1 + x}, t\_2 - \frac{t\_2}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma t z (- x))) (t_2 (/ x (+ 1.0 x))))
   (if (<= t -7.6e+28)
     (/ (+ (/ y t) x) (+ 1.0 x))
     (fma (/ y t_1) (/ z (+ 1.0 x)) (- t_2 (/ t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, z, -x);
	double t_2 = x / (1.0 + x);
	double tmp;
	if (t <= -7.6e+28) {
		tmp = ((y / t) + x) / (1.0 + x);
	} else {
		tmp = fma((y / t_1), (z / (1.0 + x)), (t_2 - (t_2 / t_1)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(t, z, Float64(-x))
	t_2 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (t <= -7.6e+28)
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(1.0 + x));
	else
		tmp = fma(Float64(y / t_1), Float64(z / Float64(1.0 + x)), Float64(t_2 - Float64(t_2 / t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.6e+28], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / t$95$1), $MachinePrecision] * N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.5999999999999998e28

   1. Initial program 78.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   if -7.5999999999999998e28 < t

   1. Initial program 89.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 94.1%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(t, z, -x\right)}, \frac{z}{1 + x}, \frac{x}{1 + x} - \frac{\frac{x}{1 + x}}{\mathsf{fma}\left(t, z, -x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x}\\ \mathbf{if}\;t\_1 \leq 10^{+299}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t} + \frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x))))
   (if (<= t_1 1e+299) t_1 (+ (/ y (* (+ 1.0 x) t)) (/ x (+ 1.0 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= 1e+299) {
		tmp = t_1;
	} else {
		tmp = (y / ((1.0 + x) * 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 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0d0 + x)
    if (t_1 <= 1d+299) then
        tmp = t_1
    else
        tmp = (y / ((1.0d0 + x) * 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 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	double tmp;
	if (t_1 <= 1e+299) {
		tmp = t_1;
	} else {
		tmp = (y / ((1.0 + x) * t)) + (x / (1.0 + x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x)
	tmp = 0
	if t_1 <= 1e+299:
		tmp = t_1
	else:
		tmp = (y / ((1.0 + x) * t)) + (x / (1.0 + x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= 1e+299)
		tmp = t_1;
	else
		tmp = Float64(Float64(y / Float64(Float64(1.0 + x) * t)) + Float64(x / Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= 1e+299)
		tmp = t_1;
	else
		tmp = (y / ((1.0 + x) * t)) + (x / (1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+299], t$95$1, N[(N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + x), $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))) < 1.0000000000000001e299

   1. Initial program 95.6%

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

   if 1.0000000000000001e299 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 19.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 76.9%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 10^{+299}:\\ \;\;\;\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot t} + \frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - \frac{x - z \cdot y}{z \cdot t - x}}{1 + x} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (- x (/ (- x (* z y)) (- (* z t) x))) (+ 1.0 x)) 2e-15)
   (* (- 1.0 x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x)) <= 2e-15) {
		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 (((x - ((x - (z * y)) / ((z * t) - x))) / (1.0d0 + x)) <= 2d-15) 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 (((x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x)) <= 2e-15) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x)) <= 2e-15:
		tmp = (1.0 - x) * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(Float64(x - Float64(z * y)) / Float64(Float64(z * t) - x))) / Float64(1.0 + x)) <= 2e-15)
		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 (((x - ((x - (z * y)) / ((z * t) - x))) / (1.0 + x)) <= 2e-15)
		tmp = (1.0 - x) * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x - N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 2e-15], 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))) < 2.0000000000000002e-15

   1. Initial program 87.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 33.5

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

   5. Applied rewrites 33.5%

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

   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 30.0%

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

   if 2.0000000000000002e-15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 86.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 75.6%

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

4. Final simplification 61.2%

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

Alternative 12: 69.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{z}{x \cdot x} \cdot y\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (* (/ z (* x x)) y))))
   (if (<= x -6.8e-15) t_1 (if (<= x 2e-56) (/ y t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - ((z / (x * x)) * y);
	double tmp;
	if (x <= -6.8e-15) {
		tmp = t_1;
	} else if (x <= 2e-56) {
		tmp = y / t;
	} 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) :: tmp
    t_1 = 1.0d0 - ((z / (x * x)) * y)
    if (x <= (-6.8d-15)) then
        tmp = t_1
    else if (x <= 2d-56) then
        tmp = y / t
    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 = 1.0 - ((z / (x * x)) * y);
	double tmp;
	if (x <= -6.8e-15) {
		tmp = t_1;
	} else if (x <= 2e-56) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - ((z / (x * x)) * y)
	tmp = 0
	if x <= -6.8e-15:
		tmp = t_1
	elif x <= 2e-56:
		tmp = y / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(Float64(z / Float64(x * x)) * y))
	tmp = 0.0
	if (x <= -6.8e-15)
		tmp = t_1;
	elseif (x <= 2e-56)
		tmp = Float64(y / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - ((z / (x * x)) * y);
	tmp = 0.0;
	if (x <= -6.8e-15)
		tmp = t_1;
	elseif (x <= 2e-56)
		tmp = y / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(N[(z / N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.8e-15], t$95$1, If[LessEqual[x, 2e-56], N[(y / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8000000000000001e-15 or 2.0000000000000001e-56 < x

   1. Initial program 84.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z - t \cdot z}{{x}^{2}}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{y \cdot z - t \cdot z}{{x}^{2}}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{y \cdot z - t \cdot z}{{x}^{2}}} \]

      4. div-sub N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{y \cdot z}{{x}^{2}} - \frac{t \cdot z}{{x}^{2}}\right)} \]

      5. associate-/l* N/A

         \[\leadsto 1 - \left(\color{blue}{y \cdot \frac{z}{{x}^{2}}} - \frac{t \cdot z}{{x}^{2}}\right) \]

      6. associate-/l* N/A

         \[\leadsto 1 - \left(y \cdot \frac{z}{{x}^{2}} - \color{blue}{t \cdot \frac{z}{{x}^{2}}}\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto 1 - \color{blue}{\frac{z}{{x}^{2}} \cdot \left(y - t\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{z}{{x}^{2}} \cdot \left(y - t\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{z}{{x}^{2}}} \cdot \left(y - t\right) \]

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 88.9%

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

   if -6.8000000000000001e-15 < x < 2.0000000000000001e-56

   1. Initial program 89.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 49.9

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

   5. Applied rewrites 49.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-15}:\\ \;\;\;\;1 - \frac{z}{x \cdot x} \cdot y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{z}{x \cdot x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.6% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 53.7%

   \[\leadsto \color{blue}{1} \]
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 2024304 
(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)))