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

Percentage Accurate: 85.1% → 89.1%

Time: 9.7s

Alternatives: 11

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 93.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 93.5

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

   4. Applied rewrites 93.5%

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

   if +inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.7%

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

Alternative 2: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (z * y)) / (t - (a * z));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (z * y)) / (t - (a * z))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (z * y)) / (t - (a * z));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 93.5%

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

   if +inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.7%

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

Alternative 3: 51.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{-t}\\ \mathbf{if}\;z \leq -5.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+109}:\\ \;\;\;\;\frac{-x}{a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* z y) (- t))))
   (if (<= z -5.3e+51)
     (/ y a)
     (if (<= z -8.5e-136)
       t_1
       (if (<= z 4.1e-112)
         (/ x t)
         (if (<= z 3.9e-41)
           t_1
           (if (<= z 7e+109) (/ (- x) (* a z)) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * y) / -t;
	double tmp;
	if (z <= -5.3e+51) {
		tmp = y / a;
	} else if (z <= -8.5e-136) {
		tmp = t_1;
	} else if (z <= 4.1e-112) {
		tmp = x / t;
	} else if (z <= 3.9e-41) {
		tmp = t_1;
	} else if (z <= 7e+109) {
		tmp = -x / (a * z);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * y) / -t
    if (z <= (-5.3d+51)) then
        tmp = y / a
    else if (z <= (-8.5d-136)) then
        tmp = t_1
    else if (z <= 4.1d-112) then
        tmp = x / t
    else if (z <= 3.9d-41) then
        tmp = t_1
    else if (z <= 7d+109) then
        tmp = -x / (a * z)
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * y) / -t;
	double tmp;
	if (z <= -5.3e+51) {
		tmp = y / a;
	} else if (z <= -8.5e-136) {
		tmp = t_1;
	} else if (z <= 4.1e-112) {
		tmp = x / t;
	} else if (z <= 3.9e-41) {
		tmp = t_1;
	} else if (z <= 7e+109) {
		tmp = -x / (a * z);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * y) / -t
	tmp = 0
	if z <= -5.3e+51:
		tmp = y / a
	elif z <= -8.5e-136:
		tmp = t_1
	elif z <= 4.1e-112:
		tmp = x / t
	elif z <= 3.9e-41:
		tmp = t_1
	elif z <= 7e+109:
		tmp = -x / (a * z)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * y) / Float64(-t))
	tmp = 0.0
	if (z <= -5.3e+51)
		tmp = Float64(y / a);
	elseif (z <= -8.5e-136)
		tmp = t_1;
	elseif (z <= 4.1e-112)
		tmp = Float64(x / t);
	elseif (z <= 3.9e-41)
		tmp = t_1;
	elseif (z <= 7e+109)
		tmp = Float64(Float64(-x) / Float64(a * z));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * y) / -t;
	tmp = 0.0;
	if (z <= -5.3e+51)
		tmp = y / a;
	elseif (z <= -8.5e-136)
		tmp = t_1;
	elseif (z <= 4.1e-112)
		tmp = x / t;
	elseif (z <= 3.9e-41)
		tmp = t_1;
	elseif (z <= 7e+109)
		tmp = -x / (a * z);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / (-t)), $MachinePrecision]}, If[LessEqual[z, -5.3e+51], N[(y / a), $MachinePrecision], If[LessEqual[z, -8.5e-136], t$95$1, If[LessEqual[z, 4.1e-112], N[(x / t), $MachinePrecision], If[LessEqual[z, 3.9e-41], t$95$1, If[LessEqual[z, 7e+109], N[((-x) / N[(a * z), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.2999999999999997e51 or 6.99999999999999966e109 < z

   1. Initial program 75.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if -5.2999999999999997e51 < z < -8.49999999999999973e-136 or 4.09999999999999996e-112 < z < 3.89999999999999991e-41

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. associate-*r* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 49.1%

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

   if -8.49999999999999973e-136 < z < 4.09999999999999996e-112

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if 3.89999999999999991e-41 < z < 6.99999999999999966e109

   1. Initial program 92.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 15.2

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

   5. Applied rewrites 15.2%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. neg-mul-1 N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 45.2%

       \[\leadsto \frac{-x}{\color{blue}{a} \cdot z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 51.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-y}{t} \cdot z\\ \mathbf{if}\;z \leq -5.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+109}:\\ \;\;\;\;\frac{-x}{a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- y) t) z)))
   (if (<= z -5.3e+51)
     (/ y a)
     (if (<= z -8.5e-136)
       t_1
       (if (<= z 4.1e-112)
         (/ x t)
         (if (<= z 3.9e-41)
           t_1
           (if (<= z 7e+109) (/ (- x) (* a z)) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-y / t) * z;
	double tmp;
	if (z <= -5.3e+51) {
		tmp = y / a;
	} else if (z <= -8.5e-136) {
		tmp = t_1;
	} else if (z <= 4.1e-112) {
		tmp = x / t;
	} else if (z <= 3.9e-41) {
		tmp = t_1;
	} else if (z <= 7e+109) {
		tmp = -x / (a * z);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-y / t) * z
    if (z <= (-5.3d+51)) then
        tmp = y / a
    else if (z <= (-8.5d-136)) then
        tmp = t_1
    else if (z <= 4.1d-112) then
        tmp = x / t
    else if (z <= 3.9d-41) then
        tmp = t_1
    else if (z <= 7d+109) then
        tmp = -x / (a * z)
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (-y / t) * z;
	double tmp;
	if (z <= -5.3e+51) {
		tmp = y / a;
	} else if (z <= -8.5e-136) {
		tmp = t_1;
	} else if (z <= 4.1e-112) {
		tmp = x / t;
	} else if (z <= 3.9e-41) {
		tmp = t_1;
	} else if (z <= 7e+109) {
		tmp = -x / (a * z);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (-y / t) * z
	tmp = 0
	if z <= -5.3e+51:
		tmp = y / a
	elif z <= -8.5e-136:
		tmp = t_1
	elif z <= 4.1e-112:
		tmp = x / t
	elif z <= 3.9e-41:
		tmp = t_1
	elif z <= 7e+109:
		tmp = -x / (a * z)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-y) / t) * z)
	tmp = 0.0
	if (z <= -5.3e+51)
		tmp = Float64(y / a);
	elseif (z <= -8.5e-136)
		tmp = t_1;
	elseif (z <= 4.1e-112)
		tmp = Float64(x / t);
	elseif (z <= 3.9e-41)
		tmp = t_1;
	elseif (z <= 7e+109)
		tmp = Float64(Float64(-x) / Float64(a * z));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (-y / t) * z;
	tmp = 0.0;
	if (z <= -5.3e+51)
		tmp = y / a;
	elseif (z <= -8.5e-136)
		tmp = t_1;
	elseif (z <= 4.1e-112)
		tmp = x / t;
	elseif (z <= 3.9e-41)
		tmp = t_1;
	elseif (z <= 7e+109)
		tmp = -x / (a * z);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-y) / t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -5.3e+51], N[(y / a), $MachinePrecision], If[LessEqual[z, -8.5e-136], t$95$1, If[LessEqual[z, 4.1e-112], N[(x / t), $MachinePrecision], If[LessEqual[z, 3.9e-41], t$95$1, If[LessEqual[z, 7e+109], N[((-x) / N[(a * z), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.2999999999999997e51 or 6.99999999999999966e109 < z

   1. Initial program 75.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if -5.2999999999999997e51 < z < -8.49999999999999973e-136 or 4.09999999999999996e-112 < z < 3.89999999999999991e-41

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. associate-*r* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.7%

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

   if -8.49999999999999973e-136 < z < 4.09999999999999996e-112

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if 3.89999999999999991e-41 < z < 6.99999999999999966e109

   1. Initial program 92.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 15.2

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

   5. Applied rewrites 15.2%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. neg-mul-1 N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 45.2%

       \[\leadsto \frac{-x}{\color{blue}{a} \cdot z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+117}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e+117)
   (/ y a)
   (if (<= z 8.8e-42)
     (/ (- x (* z y)) t)
     (if (<= z 7e+109) (/ x (fma (- z) a t)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+117) {
		tmp = y / a;
	} else if (z <= 8.8e-42) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 7e+109) {
		tmp = x / fma(-z, a, t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e+117)
		tmp = Float64(y / a);
	elseif (z <= 8.8e-42)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 7e+109)
		tmp = Float64(x / fma(Float64(-z), a, t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e+117], N[(y / a), $MachinePrecision], If[LessEqual[z, 8.8e-42], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 7e+109], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000005e117 or 6.99999999999999966e109 < z

   1. Initial program 71.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if -1.00000000000000005e117 < z < 8.8000000000000002e-42

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if 8.8000000000000002e-42 < z < 6.99999999999999966e109

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 53.5

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

   5. Applied rewrites 53.5%

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

Alternative 6: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(a, z, -t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (fma (- z) a t))))
   (if (<= x -1.4e-35)
     t_1
     (if (<= x 2.25e+61) (/ (* z y) (fma a z (- t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / fma(-z, a, t);
	double tmp;
	if (x <= -1.4e-35) {
		tmp = t_1;
	} else if (x <= 2.25e+61) {
		tmp = (z * y) / fma(a, z, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / fma(Float64(-z), a, t))
	tmp = 0.0
	if (x <= -1.4e-35)
		tmp = t_1;
	elseif (x <= 2.25e+61)
		tmp = Float64(Float64(z * y) / fma(a, z, Float64(-t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e-35], t$95$1, If[LessEqual[x, 2.25e+61], N[(N[(z * y), $MachinePrecision] / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e-35 or 2.25e61 < x

   1. Initial program 88.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -1.4e-35 < x < 2.25e61

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. associate-*r* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 78.9

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

   5. Applied rewrites 78.9%

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

Alternative 7: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (fma (- z) a t))))
   (if (<= x -1.5e-35)
     t_1
     (if (<= x 2.9e+62) (* (/ z (fma a z (- t))) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / fma(-z, a, t);
	double tmp;
	if (x <= -1.5e-35) {
		tmp = t_1;
	} else if (x <= 2.9e+62) {
		tmp = (z / fma(a, z, -t)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / fma(Float64(-z), a, t))
	tmp = 0.0
	if (x <= -1.5e-35)
		tmp = t_1;
	elseif (x <= 2.9e+62)
		tmp = Float64(Float64(z / fma(a, z, Float64(-t))) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5e-35], t$95$1, If[LessEqual[x, 2.9e+62], N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999994e-35 or 2.89999999999999984e62 < x

   1. Initial program 88.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -1.49999999999999994e-35 < x < 2.89999999999999984e62

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. associate-*r* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 78.9

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

   5. Applied rewrites 78.9%

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

   7. Applied rewrites 74.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+142}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.55e+142)
   (/ y a)
   (if (<= z 7e+109) (/ x (fma (- z) a t)) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e+142) {
		tmp = y / a;
	} else if (z <= 7e+109) {
		tmp = x / fma(-z, a, t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.55e+142)
		tmp = Float64(y / a);
	elseif (z <= 7e+109)
		tmp = Float64(x / fma(Float64(-z), a, t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.55e+142], N[(y / a), $MachinePrecision], If[LessEqual[z, 7e+109], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55e142 or 6.99999999999999966e109 < z

   1. Initial program 71.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -1.55e142 < z < 6.99999999999999966e109

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 56.8

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

   5. Applied rewrites 56.8%

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

Alternative 9: 52.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{-y}{t} \cdot z\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.3e+51)
   (/ y a)
   (if (<= z -8.5e-136)
     (* (/ (- y) t) z)
     (if (<= z 1.65e-19) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.3e+51) {
		tmp = y / a;
	} else if (z <= -8.5e-136) {
		tmp = (-y / t) * z;
	} else if (z <= 1.65e-19) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.3d+51)) then
        tmp = y / a
    else if (z <= (-8.5d-136)) then
        tmp = (-y / t) * z
    else if (z <= 1.65d-19) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.3e+51) {
		tmp = y / a;
	} else if (z <= -8.5e-136) {
		tmp = (-y / t) * z;
	} else if (z <= 1.65e-19) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.3e+51:
		tmp = y / a
	elif z <= -8.5e-136:
		tmp = (-y / t) * z
	elif z <= 1.65e-19:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.3e+51)
		tmp = Float64(y / a);
	elseif (z <= -8.5e-136)
		tmp = Float64(Float64(Float64(-y) / t) * z);
	elseif (z <= 1.65e-19)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.3e+51)
		tmp = y / a;
	elseif (z <= -8.5e-136)
		tmp = (-y / t) * z;
	elseif (z <= 1.65e-19)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.3e+51], N[(y / a), $MachinePrecision], If[LessEqual[z, -8.5e-136], N[(N[((-y) / t), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 1.65e-19], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2999999999999997e51 or 1.6499999999999999e-19 < z

   1. Initial program 80.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 55.2

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

   5. Applied rewrites 55.2%

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

   if -5.2999999999999997e51 < z < -8.49999999999999973e-136

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. associate-*r* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 48.1%

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

   if -8.49999999999999973e-136 < z < 1.6499999999999999e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 58.3

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

   5. Applied rewrites 58.3%

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

Alternative 10: 53.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.02e-91) (/ y a) (if (<= z 1.65e-19) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e-91) {
		tmp = y / a;
	} else if (z <= 1.65e-19) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.02d-91)) then
        tmp = y / a
    else if (z <= 1.65d-19) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e-91) {
		tmp = y / a;
	} else if (z <= 1.65e-19) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.02e-91:
		tmp = y / a
	elif z <= 1.65e-19:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e-91)
		tmp = Float64(y / a);
	elseif (z <= 1.65e-19)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.02e-91)
		tmp = y / a;
	elseif (z <= 1.65e-19)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e-91], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.65e-19], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.01999999999999994e-91 or 1.6499999999999999e-19 < z

   1. Initial program 85.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 50.1

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

   5. Applied rewrites 50.1%

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

   if -1.01999999999999994e-91 < z < 1.6499999999999999e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 55.9

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

   5. Applied rewrites 55.9%

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

Alternative 11: 35.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{t} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ x t))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x / t), $MachinePrecision]

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 30.8

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

5. Applied rewrites 30.8%

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

Developer Target 1: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (- (/ x t_1) (/ y (- (/ t z) a)))))
   (if (< z -32113435955957344.0)
     t_2
     (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x / t_1) - (y / ((t / z) - a))
	tmp = 0
	if z < -32113435955957344.0:
		tmp = t_2
	elif z < 3.5139522372978296e-86:
		tmp = (x - (y * z)) * (1.0 / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(Float64(t / z) - a)))
	tmp = 0.0
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = Float64(Float64(x - Float64(y * z)) * Float64(1.0 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024240 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 4392440296622287/125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))))))

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