Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B

Percentage Accurate: 95.9% → 96.5%

Time: 8.7s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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 * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-z, \frac{t}{x\_m}, \frac{y}{x\_m}\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.4e-22)
    (/ x_m (fma (- z) t y))
    (/ 1.0 (fma (- z) (/ t x_m) (/ y x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2.4e-22) {
		tmp = x_m / fma(-z, t, y);
	} else {
		tmp = 1.0 / fma(-z, (t / x_m), (y / x_m));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2.4e-22)
		tmp = Float64(x_m / fma(Float64(-z), t, y));
	else
		tmp = Float64(1.0 / fma(Float64(-z), Float64(t / x_m), Float64(y / x_m)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 2.4e-22], N[(x$95$m / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[((-z) * N[(t / x$95$m), $MachinePrecision] + N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.40000000000000002e-22

   1. Initial program 97.0%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 97.0

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

   4. Applied rewrites 97.0%

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

   if 2.40000000000000002e-22 < x

   1. Initial program 96.6%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 96.6

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

   4. Applied rewrites 96.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. lift--.f64 N/A

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

      11. div-sub N/A

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

      12. sub-neg N/A

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

      13. distribute-neg-in N/A

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

      14. distribute-neg-frac N/A

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

      15. distribute-frac-neg2 N/A

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

      16. frac-2neg N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-/l* N/A

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

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

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

      21. lift-neg.f64 N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 N/A

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

   6. Applied rewrites 96.5%

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

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq 10^{+268}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-z}{\frac{x\_m}{t}}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* t z) 1e+268)
    (/ x_m (fma (- z) t y))
    (/ 1.0 (/ (- z) (/ x_m t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t * z) <= 1e+268) {
		tmp = x_m / fma(-z, t, y);
	} else {
		tmp = 1.0 / (-z / (x_m / t));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(t * z) <= 1e+268)
		tmp = Float64(x_m / fma(Float64(-z), t, y));
	else
		tmp = Float64(1.0 / Float64(Float64(-z) / Float64(x_m / t)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(t * z), $MachinePrecision], 1e+268], N[(x$95$m / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[((-z) / N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 9.9999999999999997e267

   1. Initial program 99.0%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if 9.9999999999999997e267 < (*.f64 z t)

   1. Initial program 74.5%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 74.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 74.5

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.7

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

   10. Applied rewrites 99.7%

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

   12. Applied rewrites 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq 10^{+268}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-z}{\frac{x}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(-t\right) \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* (- t) z))))
   (* x_s (if (<= (* t z) -5e+58) t_1 (if (<= (* t z) 2e+21) (/ x_m y) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (-t * z);
	double tmp;
	if ((t * z) <= -5e+58) {
		tmp = t_1;
	} else if ((t * z) <= 2e+21) {
		tmp = x_m / y;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m / (-t * z)
    if ((t * z) <= (-5d+58)) then
        tmp = t_1
    else if ((t * z) <= 2d+21) then
        tmp = x_m / y
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (-t * z);
	double tmp;
	if ((t * z) <= -5e+58) {
		tmp = t_1;
	} else if ((t * z) <= 2e+21) {
		tmp = x_m / y;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (-t * z)
	tmp = 0
	if (t * z) <= -5e+58:
		tmp = t_1
	elif (t * z) <= 2e+21:
		tmp = x_m / y
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(Float64(-t) * z))
	tmp = 0.0
	if (Float64(t * z) <= -5e+58)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+21)
		tmp = Float64(x_m / y);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (-t * z);
	tmp = 0.0;
	if ((t * z) <= -5e+58)
		tmp = t_1;
	elseif ((t * z) <= 2e+21)
		tmp = x_m / y;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[((-t) * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(t * z), $MachinePrecision], -5e+58], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+21], N[(x$95$m / y), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999986e58 or 2e21 < (*.f64 z t)

   1. Initial program 93.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -4.99999999999999986e58 < (*.f64 z t) < 2e21

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{\left(-t\right) \cdot z}\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(-t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq 10^{+295}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-z}{x\_m} \cdot t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* t z) 1e+295)
    (/ x_m (fma (- z) t y))
    (/ 1.0 (* (/ (- z) x_m) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t * z) <= 1e+295) {
		tmp = x_m / fma(-z, t, y);
	} else {
		tmp = 1.0 / ((-z / x_m) * t);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(t * z) <= 1e+295)
		tmp = Float64(x_m / fma(Float64(-z), t, y));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(-z) / x_m) * t));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(t * z), $MachinePrecision], 1e+295], N[(x$95$m / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[((-z) / x$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 9.9999999999999998e294

   1. Initial program 99.0%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if 9.9999999999999998e294 < (*.f64 z t)

   1. Initial program 67.2%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 67.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.2

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq 10^{+295}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-z}{x} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \frac{x\_m}{\mathsf{fma}\left(-z, t, y\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (/ x_m (fma (- z) t y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / fma(-z, t, y));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / fma(Float64(-z), t, y)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

   \[\frac{x}{y - z \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 96.9

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

4. Applied rewrites 96.9%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]
5. Add Preprocessing

Alternative 6: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \frac{x\_m}{y - t \cdot z} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (/ x_m (- y (* t z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / (y - (t * z)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m / (y - (t * z)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / (y - (t * z)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	return x_s * (x_m / (y - (t * z)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / Float64(y - Float64(t * z))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / (y - (t * z)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

3. Final simplification 96.9%

   \[\leadsto \frac{x}{y - t \cdot z} \]
4. Add Preprocessing

Alternative 7: 64.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \frac{x\_m}{\mathsf{fma}\left(z, t, y\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (/ x_m (fma z t y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / fma(z, t, y));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / fma(z, t, y)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / N[(z * t + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

   \[\frac{x}{y - z \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 96.9

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

4. Applied rewrites 96.9%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. mul0-lft N/A

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

   4. lift-*.f64 N/A

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

   5. flip-- N/A

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

   6. div-inv N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. mul0-lft N/A

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

   10. lift-*.f64 N/A

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

   11. mul0-lft N/A

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

   12. metadata-eval N/A

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

   13. sub0-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. mul0-lft N/A

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

   18. +-lft-identity N/A

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

   19. lower-/.f64 81.3

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

6. Applied rewrites 81.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-neg.f64 N/A

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

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

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

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

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

   6. neg-mul-1 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 81.3

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

8. Applied rewrites 81.3%

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

10. Applied rewrites 64.3%

    \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(z, t, y\right)}} \]
11. Add Preprocessing

Alternative 8: 54.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \frac{x\_m}{y} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (/ x_m y)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / y);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m / y)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / y);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	return x_s * (x_m / y)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / y))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / y);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 52.3

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

5. Applied rewrites 52.3%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
6. Add Preprocessing

Developer Target 1: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) then
        tmp = x / (y - (z * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -161819597360704900000000000000000000000000000000000) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 213783064348764440000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t))))))

  (/ x (- y (* z t))))