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

Percentage Accurate: 85.2% → 91.5%

Time: 7.0s

Alternatives: 12

Speedup: 0.7×

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: 85.2% 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: 91.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{-x}{z}}{a} + \frac{y}{a}\\ t_2 := t - a \cdot z\\ \mathbf{if}\;z \leq -6 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+244}:\\ \;\;\;\;\frac{x}{t\_2} - \frac{y}{t\_2} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (/ (/ (- x) z) a) (/ y a))) (t_2 (- t (* a z))))
   (if (<= z -6e+116)
     t_1
     (if (<= z 1.1e+14)
       (/ (fma (- z) y x) (fma (- z) a t))
       (if (<= z 4.4e+244) (- (/ x t_2) (* (/ y t_2) z)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((-x / z) / a) + (y / a);
	double t_2 = t - (a * z);
	double tmp;
	if (z <= -6e+116) {
		tmp = t_1;
	} else if (z <= 1.1e+14) {
		tmp = fma(-z, y, x) / fma(-z, a, t);
	} else if (z <= 4.4e+244) {
		tmp = (x / t_2) - ((y / t_2) * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(Float64(-x) / z) / a) + Float64(y / a))
	t_2 = Float64(t - Float64(a * z))
	tmp = 0.0
	if (z <= -6e+116)
		tmp = t_1;
	elseif (z <= 1.1e+14)
		tmp = Float64(fma(Float64(-z), y, x) / fma(Float64(-z), a, t));
	elseif (z <= 4.4e+244)
		tmp = Float64(Float64(x / t_2) - Float64(Float64(y / t_2) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[((-x) / z), $MachinePrecision] / a), $MachinePrecision] + N[(y / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6e+116], t$95$1, If[LessEqual[z, 1.1e+14], N[(N[((-z) * y + x), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e+244], N[(N[(x / t$95$2), $MachinePrecision] - N[(N[(y / t$95$2), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.9999999999999997e116 or 4.40000000000000003e244 < z

   1. Initial program 55.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 54.2

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 82.8

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

   7. Applied rewrites 82.8%

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

   9. Applied rewrites 96.6%

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

   if -5.9999999999999997e116 < z < 1.1e14

   1. Initial program 99.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. lift-neg.f64 N/A

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

      9. lower-fma.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if 1.1e14 < z < 4.40000000000000003e244

   1. Initial program 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 90.8

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

   4. Applied rewrites 90.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{-x}{z}}{a} + \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+244}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{t - a \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{a} + \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+116} \lor \neg \left(z \leq 7 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{a} + \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6e+116) (not (<= z 7e+197)))
   (+ (/ (/ (- x) z) a) (/ y a))
   (/ (fma (- z) y x) (fma (- z) a t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6e+116) || !(z <= 7e+197)) {
		tmp = ((-x / z) / a) + (y / a);
	} else {
		tmp = fma(-z, y, x) / fma(-z, a, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6e+116) || !(z <= 7e+197))
		tmp = Float64(Float64(Float64(Float64(-x) / z) / a) + Float64(y / a));
	else
		tmp = Float64(fma(Float64(-z), y, x) / fma(Float64(-z), a, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6e+116], N[Not[LessEqual[z, 7e+197]], $MachinePrecision]], N[(N[(N[((-x) / z), $MachinePrecision] / a), $MachinePrecision] + N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[((-z) * y + x), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.9999999999999997e116 or 6.99999999999999999e197 < z

   1. Initial program 54.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 80.1

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 92.4%

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

   if -5.9999999999999997e116 < z < 6.99999999999999999e197

   1. Initial program 95.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 95.8

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

   4. Applied rewrites 95.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. lift-neg.f64 N/A

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

      9. lower-fma.f64 95.8

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

   6. Applied rewrites 95.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+116} \lor \neg \left(z \leq 7 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{a} + \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+116} \lor \neg \left(z \leq 7 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6e+116) (not (<= z 7e+197)))
   (/ (- y (/ x z)) a)
   (/ (fma (- z) y x) (fma (- z) a t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6e+116) || !(z <= 7e+197)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = fma(-z, y, x) / fma(-z, a, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6e+116) || !(z <= 7e+197))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(fma(Float64(-z), y, x) / fma(Float64(-z), a, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6e+116], N[Not[LessEqual[z, 7e+197]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[((-z) * y + x), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.9999999999999997e116 or 6.99999999999999999e197 < z

   1. Initial program 54.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in a around -inf

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

      1. associate-*r/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. lower-/.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 92.3

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

   10. Applied rewrites 92.3%

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

   if -5.9999999999999997e116 < z < 6.99999999999999999e197

   1. Initial program 95.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 95.8

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

   4. Applied rewrites 95.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. lift-neg.f64 N/A

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

      9. lower-fma.f64 95.8

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

   6. Applied rewrites 95.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+116} \lor \neg \left(z \leq 7 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+116} \lor \neg \left(z \leq 7 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6e+116) (not (<= z 7e+197)))
   (/ (- y (/ x z)) a)
   (/ (- x (* y z)) (fma (- z) a t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6e+116) || !(z <= 7e+197)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / fma(-z, a, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6e+116) || !(z <= 7e+197))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(y * z)) / fma(Float64(-z), a, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6e+116], N[Not[LessEqual[z, 7e+197]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.9999999999999997e116 or 6.99999999999999999e197 < z

   1. Initial program 54.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in a around -inf

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

      1. associate-*r/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. lower-/.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 92.3

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

   10. Applied rewrites 92.3%

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

   if -5.9999999999999997e116 < z < 6.99999999999999999e197

   1. Initial program 95.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 95.8

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

   4. Applied rewrites 95.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+116} \lor \neg \left(z \leq 7 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+116} \lor \neg \left(z \leq 7 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6e+116) (not (<= z 7e+197)))
   (/ (- y (/ x z)) a)
   (/ (- x (* y z)) (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6e+116) || !(z <= 7e+197)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / (t - (a * z));
	}
	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 <= (-6d+116)) .or. (.not. (z <= 7d+197))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (y * z)) / (t - (a * z))
    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 <= -6e+116) || !(z <= 7e+197)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -6e+116) or not (z <= 7e+197):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (y * z)) / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6e+116) || !(z <= 7e+197))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -6e+116) || ~((z <= 7e+197)))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (y * z)) / (t - (a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6e+116], N[Not[LessEqual[z, 7e+197]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.9999999999999997e116 or 6.99999999999999999e197 < z

   1. Initial program 54.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in a around -inf

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

      1. associate-*r/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. lower-/.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 92.3

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

   10. Applied rewrites 92.3%

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

   if -5.9999999999999997e116 < z < 6.99999999999999999e197

   1. Initial program 95.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+116} \lor \neg \left(z \leq 7 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8400000000 \lor \neg \left(z \leq 2.9 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -8400000000.0) (not (<= z 2.9e+137)))
   (/ (- y (/ x z)) a)
   (/ x (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -8400000000.0) || !(z <= 2.9e+137)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = x / (t - (a * z));
	}
	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 <= (-8400000000.0d0)) .or. (.not. (z <= 2.9d+137))) then
        tmp = (y - (x / z)) / a
    else
        tmp = x / (t - (a * z))
    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 <= -8400000000.0) || !(z <= 2.9e+137)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = x / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -8400000000.0) or not (z <= 2.9e+137):
		tmp = (y - (x / z)) / a
	else:
		tmp = x / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -8400000000.0) || !(z <= 2.9e+137))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(x / Float64(t - Float64(a * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -8400000000.0) || ~((z <= 2.9e+137)))
		tmp = (y - (x / z)) / a;
	else
		tmp = x / (t - (a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -8400000000.0], N[Not[LessEqual[z, 2.9e+137]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.4e9 or 2.89999999999999985e137 < z

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 76.7

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

   7. Applied rewrites 76.7%

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

   8. Taylor expanded in a around -inf

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

      1. associate-*r/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. lower-/.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 84.7

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

   10. Applied rewrites 84.7%

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

   if -8.4e9 < z < 2.89999999999999985e137

   1. Initial program 97.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8400000000 \lor \neg \left(z \leq 2.9 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-116} \lor \neg \left(x \leq 4.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a \cdot z - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= x -5.5e-116) (not (<= x 4.5e-35)))
   (/ x (- t (* a z)))
   (* y (/ z (- (* a z) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x <= -5.5e-116) || !(x <= 4.5e-35)) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y * (z / ((a * z) - t));
	}
	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 ((x <= (-5.5d-116)) .or. (.not. (x <= 4.5d-35))) then
        tmp = x / (t - (a * z))
    else
        tmp = y * (z / ((a * z) - t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x <= -5.5e-116) || !(x <= 4.5e-35)) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y * (z / ((a * z) - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x <= -5.5e-116) or not (x <= 4.5e-35):
		tmp = x / (t - (a * z))
	else:
		tmp = y * (z / ((a * z) - t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((x <= -5.5e-116) || !(x <= 4.5e-35))
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(y * Float64(z / Float64(Float64(a * z) - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x <= -5.5e-116) || ~((x <= 4.5e-35)))
		tmp = x / (t - (a * z));
	else
		tmp = y * (z / ((a * z) - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[x, -5.5e-116], N[Not[LessEqual[x, 4.5e-35]], $MachinePrecision]], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999998e-116 or 4.5000000000000001e-35 < x

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -5.4999999999999998e-116 < x < 4.5000000000000001e-35

   1. Initial program 85.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t - a \cdot z}} \]
   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. fp-cancel-sub-sign-inv N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 73.9%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-116} \lor \neg \left(x \leq 4.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a \cdot z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -10500:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+174}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -10500.0)
   (/ y a)
   (if (<= z 1.95e+52)
     (/ x t)
     (if (<= z 1.6e+174) (* (/ y t) (- z)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -10500.0) {
		tmp = y / a;
	} else if (z <= 1.95e+52) {
		tmp = x / t;
	} else if (z <= 1.6e+174) {
		tmp = (y / t) * -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) :: tmp
    if (z <= (-10500.0d0)) then
        tmp = y / a
    else if (z <= 1.95d+52) then
        tmp = x / t
    else if (z <= 1.6d+174) then
        tmp = (y / t) * -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 tmp;
	if (z <= -10500.0) {
		tmp = y / a;
	} else if (z <= 1.95e+52) {
		tmp = x / t;
	} else if (z <= 1.6e+174) {
		tmp = (y / t) * -z;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -10500.0:
		tmp = y / a
	elif z <= 1.95e+52:
		tmp = x / t
	elif z <= 1.6e+174:
		tmp = (y / t) * -z
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -10500.0)
		tmp = Float64(y / a);
	elseif (z <= 1.95e+52)
		tmp = Float64(x / t);
	elseif (z <= 1.6e+174)
		tmp = Float64(Float64(y / t) * Float64(-z));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -10500.0)
		tmp = y / a;
	elseif (z <= 1.95e+52)
		tmp = x / t;
	elseif (z <= 1.6e+174)
		tmp = (y / t) * -z;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -10500.0], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.95e+52], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.6e+174], N[(N[(y / t), $MachinePrecision] * (-z)), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -10500 or 1.6e174 < z

   1. Initial program 67.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 65.8

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

   5. Applied rewrites 65.8%

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

   if -10500 < z < 1.95e52

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 54.0

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

   5. Applied rewrites 54.0%

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

   if 1.95e52 < z < 1.6e174

   1. Initial program 69.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t - a \cdot z}} \]
   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. fp-cancel-sub-sign-inv N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-neg.f64 26.4

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

   5. Applied rewrites 26.4%

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 43.4%

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

Alternative 9: 53.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -10500:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+174}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -10500.0)
   (/ y a)
   (if (<= z 1.45e+58)
     (/ x t)
     (if (<= z 1.6e+174) (* (- y) (/ z t)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -10500.0) {
		tmp = y / a;
	} else if (z <= 1.45e+58) {
		tmp = x / t;
	} else if (z <= 1.6e+174) {
		tmp = -y * (z / 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 <= (-10500.0d0)) then
        tmp = y / a
    else if (z <= 1.45d+58) then
        tmp = x / t
    else if (z <= 1.6d+174) then
        tmp = -y * (z / 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 <= -10500.0) {
		tmp = y / a;
	} else if (z <= 1.45e+58) {
		tmp = x / t;
	} else if (z <= 1.6e+174) {
		tmp = -y * (z / t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -10500.0:
		tmp = y / a
	elif z <= 1.45e+58:
		tmp = x / t
	elif z <= 1.6e+174:
		tmp = -y * (z / t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -10500.0)
		tmp = Float64(y / a);
	elseif (z <= 1.45e+58)
		tmp = Float64(x / t);
	elseif (z <= 1.6e+174)
		tmp = Float64(Float64(-y) * Float64(z / 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 <= -10500.0)
		tmp = y / a;
	elseif (z <= 1.45e+58)
		tmp = x / t;
	elseif (z <= 1.6e+174)
		tmp = -y * (z / t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -10500.0], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.45e+58], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.6e+174], N[((-y) * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -10500 or 1.6e174 < z

   1. Initial program 67.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 65.8

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

   5. Applied rewrites 65.8%

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

   if -10500 < z < 1.45000000000000001e58

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 53.6

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

   5. Applied rewrites 53.6%

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

   if 1.45000000000000001e58 < z < 1.6e174

   1. Initial program 68.1%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]
   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 41.3

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.4%

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

Alternative 10: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+87} \lor \neg \left(z \leq 1.6 \cdot 10^{+190}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.4e+87) (not (<= z 1.6e+190))) (/ y a) (/ x (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.4e+87) || !(z <= 1.6e+190)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (a * z));
	}
	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 <= (-2.4d+87)) .or. (.not. (z <= 1.6d+190))) then
        tmp = y / a
    else
        tmp = x / (t - (a * z))
    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 <= -2.4e+87) || !(z <= 1.6e+190)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.4e+87) or not (z <= 1.6e+190):
		tmp = y / a
	else:
		tmp = x / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.4e+87) || !(z <= 1.6e+190))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(a * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.4e+87) || ~((z <= 1.6e+190)))
		tmp = y / a;
	else
		tmp = x / (t - (a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.4e+87], N[Not[LessEqual[z, 1.6e+190]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.39999999999999981e87 or 1.6e190 < z

   1. Initial program 60.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -2.39999999999999981e87 < z < 1.6e190

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+87} \lor \neg \left(z \leq 1.6 \cdot 10^{+190}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -10500 \lor \neg \left(z \leq 2.9 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -10500.0) (not (<= z 2.9e+137))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -10500.0) || !(z <= 2.9e+137)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	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 <= (-10500.0d0)) .or. (.not. (z <= 2.9d+137))) then
        tmp = y / a
    else
        tmp = x / t
    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 <= -10500.0) || !(z <= 2.9e+137)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -10500.0) or not (z <= 2.9e+137):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -10500.0) || !(z <= 2.9e+137))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -10500.0) || ~((z <= 2.9e+137)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -10500.0], N[Not[LessEqual[z, 2.9e+137]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -10500 or 2.89999999999999985e137 < z

   1. Initial program 66.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if -10500 < z < 2.89999999999999985e137

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 50.8

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

   5. Applied rewrites 50.8%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -10500 \lor \neg \left(z \leq 2.9 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.7% 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 85.2%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 35.4

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

5. Applied rewrites 35.4%

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

Developer Target 1: 97.4% 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 2024327 
(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))))