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

Percentage Accurate: 90.0% → 97.1%

Time: 9.9s

Alternatives: 18

Speedup: 0.4×

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: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;\frac{y}{t\_2 \cdot \frac{1 + x}{z}}\\ \mathbf{elif}\;t\_1 \leq 0.99995:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\left(1 + x\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_2 (fma t z (- x))))
   (if (<= t_1 -1000000000.0)
     (/ y (* t_2 (/ (+ 1.0 x) z)))
     (if (<= t_1 0.99995)
       (/ (- x (/ (- (/ x z) y) t)) (+ x 1.0))
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_1 INFINITY)
           (* y (/ z (* (+ 1.0 x) t_2)))
           (/ (+ (/ y t) x) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_2 = fma(t, z, -x);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = y / (t_2 * ((1.0 + x) / z));
	} else if (t_1 <= 0.99995) {
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0);
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y * (z / ((1.0 + x) * t_2));
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_2 = fma(t, z, Float64(-x))
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = Float64(y / Float64(t_2 * Float64(Float64(1.0 + x) / z)));
	elseif (t_1 <= 0.99995)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x + 1.0));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_1 <= Inf)
		tmp = Float64(y * Float64(z / Float64(Float64(1.0 + x) * t_2)));
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000.0], N[(y / N[(t$95$2 * N[(N[(1.0 + x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.99995], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(y * N[(z / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 67.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 85.5

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 90.2%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999950000000000006

   1. Initial program 93.6%

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

   3. Taylor expanded in t around -inf

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 0.999950000000000006 < (/.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 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 86.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 96.3%

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

   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 5 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq -1000000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, z, -x\right) \cdot \frac{1 + x}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 0.99995:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\left(1 + x\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;\frac{y}{t\_2} \cdot \frac{z}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 0.99995:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\left(1 + x\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_2 (fma t z (- x))))
   (if (<= t_1 -1000000000.0)
     (* (/ y t_2) (/ z (+ 1.0 x)))
     (if (<= t_1 0.99995)
       (/ (- x (/ (- (/ x z) y) t)) (+ x 1.0))
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_1 INFINITY)
           (* y (/ z (* (+ 1.0 x) t_2)))
           (/ (+ (/ y t) x) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_2 = fma(t, z, -x);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = (y / t_2) * (z / (1.0 + x));
	} else if (t_1 <= 0.99995) {
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0);
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y * (z / ((1.0 + x) * t_2));
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_2 = fma(t, z, Float64(-x))
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = Float64(Float64(y / t_2) * Float64(z / Float64(1.0 + x)));
	elseif (t_1 <= 0.99995)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x + 1.0));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_1 <= Inf)
		tmp = Float64(y * Float64(z / Float64(Float64(1.0 + x) * t_2)));
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000.0], N[(N[(y / t$95$2), $MachinePrecision] * N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.99995], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(y * N[(z / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 67.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999950000000000006

   1. Initial program 93.6%

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

   3. Taylor expanded in t around -inf

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \frac{-1 \cdot y - -1 \cdot \frac{x}{z}}{t}}}{x + 1} \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 0.999950000000000006 < (/.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 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 86.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 96.3%

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

   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 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_3 := \mathsf{fma}\left(t, z, -x\right)\\ \mathbf{if}\;t\_2 \leq -1000000000:\\ \;\;\;\;\frac{y}{t\_3} \cdot \frac{z}{1 + x}\\ \mathbf{elif}\;t\_2 \leq 0.99995:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_3}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\left(1 + x\right) \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_3 (fma t z (- x))))
   (if (<= t_2 -1000000000.0)
     (* (/ y t_3) (/ z (+ 1.0 x)))
     (if (<= t_2 0.99995)
       t_1
       (if (<= t_2 2.0)
         (/ (- x (/ x t_3)) (+ x 1.0))
         (if (<= t_2 INFINITY) (* y (/ z (* (+ 1.0 x) t_3))) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_3 = fma(t, z, -x);
	double tmp;
	if (t_2 <= -1000000000.0) {
		tmp = (y / t_3) * (z / (1.0 + x));
	} else if (t_2 <= 0.99995) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_3)) / (x + 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = y * (z / ((1.0 + x) * t_3));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_3 = fma(t, z, Float64(-x))
	tmp = 0.0
	if (t_2 <= -1000000000.0)
		tmp = Float64(Float64(y / t_3) * Float64(z / Float64(1.0 + x)));
	elseif (t_2 <= 0.99995)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_3)) / Float64(x + 1.0));
	elseif (t_2 <= Inf)
		tmp = Float64(y * Float64(z / Float64(Float64(1.0 + x) * t_3)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(t * z + (-x)), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000000.0], N[(N[(y / t$95$3), $MachinePrecision] * N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99995], t$95$1, If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(y * N[(z / N[(N[(1.0 + x), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 67.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999950000000000006 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   if 0.999950000000000006 < (/.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 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 86.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 96.3%

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

Alternative 4: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := y \cdot \frac{z}{\left(1 + x\right) \cdot t\_2}\\ t_4 := \frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.99995:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_2 (fma t z (- x)))
        (t_3 (* y (/ z (* (+ 1.0 x) t_2))))
        (t_4 (/ (+ (/ y t) x) (+ x 1.0))))
   (if (<= t_1 -1000000000.0)
     t_3
     (if (<= t_1 0.99995)
       t_4
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_1 INFINITY) t_3 t_4))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_2 = fma(t, z, -x);
	double t_3 = y * (z / ((1.0 + x) * t_2));
	double t_4 = ((y / t) + x) / (x + 1.0);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_3;
	} else if (t_1 <= 0.99995) {
		tmp = t_4;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(y * Float64(z / Float64(Float64(1.0 + x) * t_2)))
	t_4 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = t_3;
	elseif (t_1 <= 0.99995)
		tmp = t_4;
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(z / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000.0], t$95$3, If[LessEqual[t$95$1, 0.99995], t$95$4, If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

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

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   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 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 89.5%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999950000000000006 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   if 0.999950000000000006 < (/.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%

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

Alternative 5: 93.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{\left(1 + x\right) \cdot \mathsf{fma}\left(t, z, -x\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_3 := \frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.99995:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ z (* (+ 1.0 x) (fma t z (- x))))))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_3 (/ (+ (/ y t) x) (+ x 1.0))))
   (if (<= t_2 -1000000000.0)
     t_1
     (if (<= t_2 0.99995)
       t_3
       (if (<= t_2 2.0) 1.0 (if (<= t_2 INFINITY) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (z / ((1.0 + x) * fma(t, z, -x)));
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_3 = ((y / t) + x) / (x + 1.0);
	double tmp;
	if (t_2 <= -1000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.99995) {
		tmp = t_3;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(z / Float64(Float64(1.0 + x) * fma(t, z, Float64(-x)))))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_3 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1000000000.0)
		tmp = t_1;
	elseif (t_2 <= 0.99995)
		tmp = t_3;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / N[(N[(1.0 + x), $MachinePrecision] * N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000000.0], t$95$1, If[LessEqual[t$95$2, 0.99995], t$95$3, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

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

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   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 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 89.5%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999950000000000006 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   if 0.999950000000000006 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

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

Alternative 6: 82.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(t, x, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+34}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot \left(-y\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma t x t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 -2e+97)
     t_1
     (if (<= t_2 -2e+34)
       (* (/ z (fma x x x)) (- y))
       (if (<= t_2 1e-11) (/ (+ (/ y t) x) 1.0) (if (<= t_2 2.0) 1.0 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(t, x, t);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+97) {
		tmp = t_1;
	} else if (t_2 <= -2e+34) {
		tmp = (z / fma(x, x, x)) * -y;
	} else if (t_2 <= 1e-11) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(t, x, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e+97)
		tmp = t_1;
	elseif (t_2 <= -2e+34)
		tmp = Float64(Float64(z / fma(x, x, x)) * Float64(-y));
	elseif (t_2 <= 1e-11)
		tmp = Float64(Float64(Float64(y / t) + x) / 1.0);
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -2e+97], t$95$1, If[LessEqual[t$95$2, -2e+34], N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t$95$2, 1e-11], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.0000000000000001e97 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 59.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 26.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 59.6%

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

   if -2.0000000000000001e97 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999989e34

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 85.2%

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

   10. Applied rewrites 85.9%

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

   if -1.99999999999999989e34 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999939e-12

   1. Initial program 93.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   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 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.0%

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

   if 9.99999999999999939e-12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.6%

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

Alternative 7: 76.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(t, x, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1000000000:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(x, x, x\right)} \cdot \left(-y\right)\\ \mathbf{elif}\;t\_2 \leq 0.996:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma t x t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 -2e+97)
     t_1
     (if (<= t_2 -1000000000.0)
       (* (/ z (fma x x x)) (- y))
       (if (<= t_2 0.996) (/ x (+ 1.0 x)) (if (<= t_2 2.0) 1.0 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(t, x, t);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+97) {
		tmp = t_1;
	} else if (t_2 <= -1000000000.0) {
		tmp = (z / fma(x, x, x)) * -y;
	} else if (t_2 <= 0.996) {
		tmp = x / (1.0 + x);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(t, x, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e+97)
		tmp = t_1;
	elseif (t_2 <= -1000000000.0)
		tmp = Float64(Float64(z / fma(x, x, x)) * Float64(-y));
	elseif (t_2 <= 0.996)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -2e+97], t$95$1, If[LessEqual[t$95$2, -1000000000.0], N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t$95$2, 0.996], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.0000000000000001e97 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 59.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 26.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 59.6%

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

   if -2.0000000000000001e97 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e9

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-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 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 74.0%

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

   10. Applied rewrites 74.5%

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

   if -1e9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.996

   1. Initial program 93.5%

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

   3. Taylor expanded in 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 60.0

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

   5. Applied rewrites 60.0%

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

   if 0.996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.1%

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

Alternative 8: 88.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 0.99995:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{y \cdot z}{1 \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 0.99995)
     t_1
     (if (<= t_2 2.0)
       1.0
       (if (<= t_2 2e+263) (/ (* y z) (* 1.0 (fma t z (- x)))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.99995) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= 2e+263) {
		tmp = (y * z) / (1.0 * fma(t, z, -x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 0.99995)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= 2e+263)
		tmp = Float64(Float64(y * z) / Float64(1.0 * fma(t, z, Float64(-x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, 0.99995], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, 2e+263], N[(N[(y * z), $MachinePrecision] / N[(1.0 * N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999950000000000006 or 2.00000000000000003e263 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   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 76.1

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

   5. Applied rewrites 76.1%

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

   if 0.999950000000000006 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-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 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 89.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 0.99995:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\frac{y \cdot z}{1 \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 0.99995:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{1 \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 0.99995)
     t_1
     (if (<= t_2 2.0)
       1.0
       (if (<= t_2 INFINITY) (* z (/ y (* 1.0 (fma t z (- x))))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.99995) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = z * (y / (1.0 * fma(t, z, -x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 0.99995)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	elseif (t_2 <= Inf)
		tmp = Float64(z * Float64(y / Float64(1.0 * fma(t, z, Float64(-x)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, 0.99995], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, If[LessEqual[t$95$2, Infinity], N[(z * N[(y / N[(1.0 * N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999950000000000006 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.4%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if 0.999950000000000006 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 86.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.7%

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

   12. Applied rewrites 69.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 0.99995:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{1 \cdot \mathsf{fma}\left(t, z, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\mathsf{fma}\left(t, x, t\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -200000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.996:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (fma t x t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 -200000000000.0)
     t_1
     (if (<= t_2 0.996) (/ x (+ 1.0 x)) (if (<= t_2 2.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / fma(t, x, t);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -200000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.996) {
		tmp = x / (1.0 + x);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / fma(t, x, t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -200000000000.0)
		tmp = t_1;
	elseif (t_2 <= 0.996)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(t * x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -200000000000.0], t$95$1, If[LessEqual[t$95$2, 0.996], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e11 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 62.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 30.8%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 56.4%

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

   if -2e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.996

   1. Initial program 93.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if 0.996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.1%

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

Alternative 11: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -200000000000:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.996:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 -200000000000.0)
     (/ y t)
     (if (<= t_1 0.996) (/ x (+ 1.0 x)) (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -200000000000.0) {
		tmp = y / t;
	} else if (t_1 <= 0.996) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_1 <= (-200000000000.0d0)) then
        tmp = y / t
    else if (t_1 <= 0.996d0) then
        tmp = x / (1.0d0 + x)
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -200000000000.0) {
		tmp = y / t;
	} else if (t_1 <= 0.996) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -200000000000.0:
		tmp = y / t
	elif t_1 <= 0.996:
		tmp = x / (1.0 + x)
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -200000000000.0)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.996)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -200000000000.0)
		tmp = y / t;
	elseif (t_1 <= 0.996)
		tmp = x / (1.0 + x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200000000000.0], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.996], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e11 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 62.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 50.2

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

   5. Applied rewrites 50.2%

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

   if -2e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.996

   1. Initial program 93.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if 0.996 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.1%

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

Alternative 12: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-257}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 10^{-11}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 5e-257)
     (/ y t)
     (if (<= t_1 1e-11) (* (- 1.0 x) x) (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e-257) {
		tmp = y / t;
	} else if (t_1 <= 1e-11) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_1 <= 5d-257) then
        tmp = y / t
    else if (t_1 <= 1d-11) then
        tmp = (1.0d0 - x) * x
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e-257) {
		tmp = y / t;
	} else if (t_1 <= 1e-11) {
		tmp = (1.0 - x) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= 5e-257:
		tmp = y / t
	elif t_1 <= 1e-11:
		tmp = (1.0 - x) * x
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= 5e-257)
		tmp = Float64(y / t);
	elseif (t_1 <= 1e-11)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= 5e-257)
		tmp = y / t;
	elseif (t_1 <= 1e-11)
		tmp = (1.0 - x) * x;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-257], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 1e-11], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999989e-257 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 47.8

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

   5. Applied rewrites 47.8%

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

   if 4.99999999999999989e-257 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999939e-12

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 80.7

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

   5. Applied rewrites 80.7%

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

   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 80.8%

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

   if 9.99999999999999939e-12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.6%

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

Alternative 13: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{z - \frac{x}{y}}{\mathsf{fma}\left(t, z, -x\right)} \cdot y + x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 -2e-52)
     (/ (+ (* (/ (- z (/ x y)) (fma t z (- x))) y) x) (+ x 1.0))
     (if (<= t_1 INFINITY) t_1 (/ (+ (/ y t) x) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e-52) {
		tmp = ((((z - (x / y)) / fma(t, z, -x)) * y) + x) / (x + 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -2e-52)
		tmp = Float64(Float64(Float64(Float64(Float64(z - Float64(x / y)) / fma(t, z, Float64(-x))) * y) + x) / Float64(x + 1.0));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-52], N[(N[(N[(N[(N[(z - N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e-52

   1. Initial program 73.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 97.8

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

   7. Applied rewrites 99.7%

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

   if -2e-52 < (/.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 96.3%

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

   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 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, z, -x\right) \cdot \frac{1 + x}{z}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 -5e+47)
     (/ y (* (fma t z (- x)) (/ (+ 1.0 x) z)))
     (if (<= t_1 INFINITY) t_1 (/ (+ (/ y t) x) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -5e+47) {
		tmp = y / (fma(t, z, -x) * ((1.0 + x) / z));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -5e+47)
		tmp = Float64(y / Float64(fma(t, z, Float64(-x)) * Float64(Float64(1.0 + x) / z)));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+47], N[(y / N[(N[(t * z + (-x)), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000022e47

   1. Initial program 61.4%

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

   3. Taylor expanded in y around inf

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

      1. *-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 84.1

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 89.7%

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

   if -5.00000000000000022e47 < (/.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 96.6%

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

   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 3 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, z, -x\right) \cdot \frac{1 + x}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 0.99995 \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (or (<= t_1 0.99995) (not (<= t_1 2.0)))
     (/ (+ (/ y t) x) (+ x 1.0))
     1.0)))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if ((t_1 <= 0.99995) || !(t_1 <= 2.0)) {
		tmp = ((y / t) + x) / (x + 1.0);
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if ((t_1 <= 0.99995d0) .or. (.not. (t_1 <= 2.0d0))) then
        tmp = ((y / t) + x) / (x + 1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if ((t_1 <= 0.99995) || !(t_1 <= 2.0)) {
		tmp = ((y / t) + x) / (x + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if (t_1 <= 0.99995) or not (t_1 <= 2.0):
		tmp = ((y / t) + x) / (x + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if ((t_1 <= 0.99995) || !(t_1 <= 2.0))
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if ((t_1 <= 0.99995) || ~((t_1 <= 2.0)))
		tmp = ((y / t) + x) / (x + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.99995], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999950000000000006 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 71.4

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

   5. Applied rewrites 71.4%

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

   if 0.999950000000000006 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 0.99995 \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 2\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 94.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * z + (-x)), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(x + N[(N[(N[(z / t$95$1), $MachinePrecision] - N[(x / N[(t$95$1 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $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))) < +inf.0

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   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 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) 1e-11)
   (* (- 1.0 x) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)) <= 1e-11) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)) <= 1d-11) then
        tmp = (1.0d0 - x) * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)) <= 1e-11) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)) <= 1e-11:
		tmp = (1.0 - x) * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0)) <= 1e-11)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)) <= 1e-11)
		tmp = (1.0 - x) * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999939e-12

   1. Initial program 81.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 35.4

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

   5. Applied rewrites 35.4%

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

   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 32.4%

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

   if 9.99999999999999939e-12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 77.2%

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

Alternative 18: 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 86.9%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 52.2%

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

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024314 
(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)))