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

Percentage Accurate: 85.6% → 96.8%

Time: 15.3s

Alternatives: 11

Speedup: 0.4×

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: 85.6% 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: 96.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{\frac{x}{z} - y}{\frac{t\_1}{z}}\\ t_3 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-319}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+284}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a)))
        (t_2 (/ (- (/ x z) y) (/ t_1 z)))
        (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 (- INFINITY))
     t_2
     (if (<= t_3 -2e-319)
       t_3
       (if (<= t_3 0.0)
         (* (/ 1.0 z) (/ (- (* y z) x) a))
         (if (<= t_3 5e+284) t_3 (if (<= t_3 INFINITY) t_2 (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = ((x / z) - y) / (t_1 / z);
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= -2e-319) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = (1.0 / z) * (((y * z) - x) / a);
	} else if (t_3 <= 5e+284) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = ((x / z) - y) / (t_1 / z);
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_3 <= -2e-319) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = (1.0 / z) * (((y * z) - x) / a);
	} else if (t_3 <= 5e+284) {
		tmp = t_3;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = ((x / z) - y) / (t_1 / z)
	t_3 = (x - (y * z)) / t_1
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_2
	elif t_3 <= -2e-319:
		tmp = t_3
	elif t_3 <= 0.0:
		tmp = (1.0 / z) * (((y * z) - x) / a)
	elif t_3 <= 5e+284:
		tmp = t_3
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(Float64(x / z) - y) / Float64(t_1 / z))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= -2e-319)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(y * z) - x) / a));
	elseif (t_3 <= 5e+284)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = ((x / z) - y) / (t_1 / z);
	t_3 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_2;
	elseif (t_3 <= -2e-319)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = (1.0 / z) * (((y * z) - x) / a);
	elseif (t_3 <= 5e+284)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$2, If[LessEqual[t$95$3, -2e-319], t$95$3, If[LessEqual[t$95$3, 0.0], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+284], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$2, N[(y / a), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or 4.9999999999999999e284 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

   3. Simplified 71.8%

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

      1. div-sub 71.8%

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

      2. associate-/l* 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. frac-sub 71.3%

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

      2. associate-/r* 71.7%

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

   8. Applied egg-rr 71.7%

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

   9. Taylor expanded in x around -inf 100.0%

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

       1. +-commutative 100.0%

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

       2. mul-1-neg 100.0%

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

       3. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.99998e-319 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.9999999999999999e284

   1. Initial program 99.7%

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

   if -1.99998e-319 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 49.8%

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

      1. *-commutative 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in t around 0 29.9%

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

      1. associate-*r/ 29.9%

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

      2. neg-mul-1 29.9%

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

      3. neg-sub0 29.9%

         \[\leadsto \frac{\color{blue}{0 - \left(x - y \cdot z\right)}}{a \cdot z} \]

      4. sub-neg 29.9%

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

      5. distribute-rgt-neg-out 29.9%

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

      6. +-commutative 29.9%

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

      7. associate--r+ 29.9%

         \[\leadsto \frac{\color{blue}{\left(0 - y \cdot \left(-z\right)\right) - x}}{a \cdot z} \]

      8. neg-sub0 29.9%

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

      9. distribute-rgt-neg-out 29.9%

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

      10. remove-double-neg 29.9%

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

      11. *-commutative 29.9%

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

   7. Simplified 29.9%

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

      1. *-un-lft-identity 29.9%

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

      2. times-frac 86.1%

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

      3. *-commutative 86.1%

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

   9. Applied egg-rr 86.1%

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

   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. Step-by-step derivation

      1. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{z} - y}{\frac{t - z \cdot a}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-319}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{\frac{x}{z} - y}{\frac{t - z \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x}{t\_1} - \frac{y}{\frac{t\_1}{z}}\\ t_3 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-319}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \mathbf{elif}\;t\_3 \leq 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -z, x\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a)))
        (t_2 (- (/ x t_1) (/ y (/ t_1 z))))
        (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 -2e-319)
     t_2
     (if (<= t_3 0.0)
       (* (/ 1.0 z) (/ (- (* y z) x) a))
       (if (<= t_3 1e-17)
         (/ (fma y (- z) x) t_1)
         (if (<= t_3 INFINITY) t_2 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x / t_1) - (y / (t_1 / z));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -2e-319) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = (1.0 / z) * (((y * z) - x) / a);
	} else if (t_3 <= 1e-17) {
		tmp = fma(y, -z, x) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(t_1 / z)))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= -2e-319)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(y * z) - x) / a));
	elseif (t_3 <= 1e-17)
		tmp = Float64(fma(y, Float64(-z), x) / t_1);
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-319], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-17], N[(N[(y * (-z) + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.99998e-319 or 1.00000000000000007e-17 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 92.7%

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

      1. *-commutative 92.7%

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

   3. Simplified 92.7%

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

      1. div-sub 92.7%

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

      2. associate-/l* 97.6%

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

   6. Applied egg-rr 97.6%

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

   if -1.99998e-319 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 49.8%

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

      1. *-commutative 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in t around 0 29.9%

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

      1. associate-*r/ 29.9%

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

      2. neg-mul-1 29.9%

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

      3. neg-sub0 29.9%

         \[\leadsto \frac{\color{blue}{0 - \left(x - y \cdot z\right)}}{a \cdot z} \]

      4. sub-neg 29.9%

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

      5. distribute-rgt-neg-out 29.9%

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

      6. +-commutative 29.9%

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

      7. associate--r+ 29.9%

         \[\leadsto \frac{\color{blue}{\left(0 - y \cdot \left(-z\right)\right) - x}}{a \cdot z} \]

      8. neg-sub0 29.9%

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

      9. distribute-rgt-neg-out 29.9%

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

      10. remove-double-neg 29.9%

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

      11. *-commutative 29.9%

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

   7. Simplified 29.9%

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

      1. *-un-lft-identity 29.9%

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

      2. times-frac 86.1%

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

      3. *-commutative 86.1%

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

   9. Applied egg-rr 86.1%

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.00000000000000007e-17

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

      4. associate-/l* 89.4%

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

      5. *-commutative 89.4%

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

      6. sub-neg 89.4%

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

      7. associate-/r/ 94.9%

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

      8. associate-*l/ 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. distribute-rgt-neg-out 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   7. Simplified 99.9%

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

   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. Step-by-step derivation

      1. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-319}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -z, x\right)}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x}{t\_1} - \frac{y}{\frac{t\_1}{z}}\\ t_3 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-319}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \mathbf{elif}\;t\_3 \leq 10^{-17}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a)))
        (t_2 (- (/ x t_1) (/ y (/ t_1 z))))
        (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 -2e-319)
     t_2
     (if (<= t_3 0.0)
       (* (/ 1.0 z) (/ (- (* y z) x) a))
       (if (<= t_3 1e-17) t_3 (if (<= t_3 INFINITY) t_2 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x / t_1) - (y / (t_1 / z));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -2e-319) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = (1.0 / z) * (((y * z) - x) / a);
	} else if (t_3 <= 1e-17) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x / t_1) - (y / (t_1 / z));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -2e-319) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = (1.0 / z) * (((y * z) - x) / a);
	} else if (t_3 <= 1e-17) {
		tmp = t_3;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = (x / t_1) - (y / (t_1 / z))
	t_3 = (x - (y * z)) / t_1
	tmp = 0
	if t_3 <= -2e-319:
		tmp = t_2
	elif t_3 <= 0.0:
		tmp = (1.0 / z) * (((y * z) - x) / a)
	elif t_3 <= 1e-17:
		tmp = t_3
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(t_1 / z)))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= -2e-319)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(y * z) - x) / a));
	elseif (t_3 <= 1e-17)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = (x / t_1) - (y / (t_1 / z));
	t_3 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_3 <= -2e-319)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = (1.0 / z) * (((y * z) - x) / a);
	elseif (t_3 <= 1e-17)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-319], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-17], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$2, N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.99998e-319 or 1.00000000000000007e-17 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 92.7%

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

      1. *-commutative 92.7%

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

   3. Simplified 92.7%

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

      1. div-sub 92.7%

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

      2. associate-/l* 97.6%

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

   6. Applied egg-rr 97.6%

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

   if -1.99998e-319 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 49.8%

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

      1. *-commutative 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in t around 0 29.9%

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

      1. associate-*r/ 29.9%

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

      2. neg-mul-1 29.9%

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

      3. neg-sub0 29.9%

         \[\leadsto \frac{\color{blue}{0 - \left(x - y \cdot z\right)}}{a \cdot z} \]

      4. sub-neg 29.9%

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

      5. distribute-rgt-neg-out 29.9%

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

      6. +-commutative 29.9%

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

      7. associate--r+ 29.9%

         \[\leadsto \frac{\color{blue}{\left(0 - y \cdot \left(-z\right)\right) - x}}{a \cdot z} \]

      8. neg-sub0 29.9%

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

      9. distribute-rgt-neg-out 29.9%

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

      10. remove-double-neg 29.9%

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

      11. *-commutative 29.9%

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

   7. Simplified 29.9%

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

      1. *-un-lft-identity 29.9%

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

      2. times-frac 86.1%

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

      3. *-commutative 86.1%

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

   9. Applied egg-rr 86.1%

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.00000000000000007e-17

   1. Initial program 99.9%

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

      1. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-319}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{-17}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ t_3 := y \cdot \frac{-z}{t\_1}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-104}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-61}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{t\_1}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+198} \lor \neg \left(z \leq 10^{+267}\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a)))
        (t_2 (/ (- y (/ x z)) a))
        (t_3 (* y (/ (- z) t_1))))
   (if (<= z -1e-104)
     t_2
     (if (<= z 6.7e-128)
       (/ (- x (* y z)) t)
       (if (<= z 4.8e-61)
         t_3
         (if (<= z 2.55e-28)
           (/ x t_1)
           (if (or (<= z 1.7e+198) (not (<= z 1e+267))) t_2 t_3)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (y - (x / z)) / a;
	double t_3 = y * (-z / t_1);
	double tmp;
	if (z <= -1e-104) {
		tmp = t_2;
	} else if (z <= 6.7e-128) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 4.8e-61) {
		tmp = t_3;
	} else if (z <= 2.55e-28) {
		tmp = x / t_1;
	} else if ((z <= 1.7e+198) || !(z <= 1e+267)) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = t - (z * a)
    t_2 = (y - (x / z)) / a
    t_3 = y * (-z / t_1)
    if (z <= (-1d-104)) then
        tmp = t_2
    else if (z <= 6.7d-128) then
        tmp = (x - (y * z)) / t
    else if (z <= 4.8d-61) then
        tmp = t_3
    else if (z <= 2.55d-28) then
        tmp = x / t_1
    else if ((z <= 1.7d+198) .or. (.not. (z <= 1d+267))) then
        tmp = t_2
    else
        tmp = t_3
    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 - (z * a);
	double t_2 = (y - (x / z)) / a;
	double t_3 = y * (-z / t_1);
	double tmp;
	if (z <= -1e-104) {
		tmp = t_2;
	} else if (z <= 6.7e-128) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 4.8e-61) {
		tmp = t_3;
	} else if (z <= 2.55e-28) {
		tmp = x / t_1;
	} else if ((z <= 1.7e+198) || !(z <= 1e+267)) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = (y - (x / z)) / a
	t_3 = y * (-z / t_1)
	tmp = 0
	if z <= -1e-104:
		tmp = t_2
	elif z <= 6.7e-128:
		tmp = (x - (y * z)) / t
	elif z <= 4.8e-61:
		tmp = t_3
	elif z <= 2.55e-28:
		tmp = x / t_1
	elif (z <= 1.7e+198) or not (z <= 1e+267):
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(y - Float64(x / z)) / a)
	t_3 = Float64(y * Float64(Float64(-z) / t_1))
	tmp = 0.0
	if (z <= -1e-104)
		tmp = t_2;
	elseif (z <= 6.7e-128)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 4.8e-61)
		tmp = t_3;
	elseif (z <= 2.55e-28)
		tmp = Float64(x / t_1);
	elseif ((z <= 1.7e+198) || !(z <= 1e+267))
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = (y - (x / z)) / a;
	t_3 = y * (-z / t_1);
	tmp = 0.0;
	if (z <= -1e-104)
		tmp = t_2;
	elseif (z <= 6.7e-128)
		tmp = (x - (y * z)) / t;
	elseif (z <= 4.8e-61)
		tmp = t_3;
	elseif (z <= 2.55e-28)
		tmp = x / t_1;
	elseif ((z <= 1.7e+198) || ~((z <= 1e+267)))
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[((-z) / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e-104], t$95$2, If[LessEqual[z, 6.7e-128], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 4.8e-61], t$95$3, If[LessEqual[z, 2.55e-28], N[(x / t$95$1), $MachinePrecision], If[Or[LessEqual[z, 1.7e+198], N[Not[LessEqual[z, 1e+267]], $MachinePrecision]], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.99999999999999927e-105 or 2.55000000000000004e-28 < z < 1.7e198 or 9.9999999999999997e266 < z

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

   3. Simplified 77.6%

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

      1. div-sub 77.6%

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

      2. associate-/l* 85.1%

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

   6. Applied egg-rr 85.1%

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

   7. Taylor expanded in t around 0 79.4%

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

      1. distribute-lft-out-- 79.4%

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

      2. *-commutative 79.4%

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

      3. associate-/r* 80.8%

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

      4. div-sub 80.8%

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

      5. mul-1-neg 80.8%

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

   9. Simplified 80.8%

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

   if -9.99999999999999927e-105 < z < 6.70000000000000027e-128

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 80.2%

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

   if 6.70000000000000027e-128 < z < 4.8000000000000002e-61 or 1.7e198 < z < 9.9999999999999997e266

   1. Initial program 90.3%

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

      1. *-commutative 90.3%

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

   3. Simplified 90.3%

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

      1. div-sub 90.3%

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

      2. associate-/l* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0 71.7%

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

      1. mul-1-neg 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-*r/ 80.8%

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

      4. distribute-rgt-neg-in 80.8%

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

      5. distribute-neg-frac 80.8%

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

      6. *-commutative 80.8%

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

   9. Simplified 80.8%

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

   if 4.8000000000000002e-61 < z < 2.55000000000000004e-28

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 81.1%

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

      1. *-commutative 81.1%

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

   7. Simplified 81.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-104}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-61}:\\ \;\;\;\;y \cdot \frac{-z}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+198} \lor \neg \left(z \leq 10^{+267}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-z}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-104}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{t\_1}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-116} \lor \neg \left(z \leq 9.5 \cdot 10^{-29}\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (/ (- y (/ x z)) a)))
   (if (<= z -1e-104)
     t_2
     (if (<= z 6.7e-128)
       (/ t_1 t)
       (if (or (<= z 4e-116) (not (<= z 9.5e-29))) t_2 (/ 1.0 (/ t t_1)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (z <= -1e-104) {
		tmp = t_2;
	} else if (z <= 6.7e-128) {
		tmp = t_1 / t;
	} else if ((z <= 4e-116) || !(z <= 9.5e-29)) {
		tmp = t_2;
	} else {
		tmp = 1.0 / (t / t_1);
	}
	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 = x - (y * z)
    t_2 = (y - (x / z)) / a
    if (z <= (-1d-104)) then
        tmp = t_2
    else if (z <= 6.7d-128) then
        tmp = t_1 / t
    else if ((z <= 4d-116) .or. (.not. (z <= 9.5d-29))) then
        tmp = t_2
    else
        tmp = 1.0d0 / (t / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (z <= -1e-104) {
		tmp = t_2;
	} else if (z <= 6.7e-128) {
		tmp = t_1 / t;
	} else if ((z <= 4e-116) || !(z <= 9.5e-29)) {
		tmp = t_2;
	} else {
		tmp = 1.0 / (t / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - (y * z)
	t_2 = (y - (x / z)) / a
	tmp = 0
	if z <= -1e-104:
		tmp = t_2
	elif z <= 6.7e-128:
		tmp = t_1 / t
	elif (z <= 4e-116) or not (z <= 9.5e-29):
		tmp = t_2
	else:
		tmp = 1.0 / (t / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -1e-104)
		tmp = t_2;
	elseif (z <= 6.7e-128)
		tmp = Float64(t_1 / t);
	elseif ((z <= 4e-116) || !(z <= 9.5e-29))
		tmp = t_2;
	else
		tmp = Float64(1.0 / Float64(t / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - (y * z);
	t_2 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -1e-104)
		tmp = t_2;
	elseif (z <= 6.7e-128)
		tmp = t_1 / t;
	elseif ((z <= 4e-116) || ~((z <= 9.5e-29)))
		tmp = t_2;
	else
		tmp = 1.0 / (t / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -1e-104], t$95$2, If[LessEqual[z, 6.7e-128], N[(t$95$1 / t), $MachinePrecision], If[Or[LessEqual[z, 4e-116], N[Not[LessEqual[z, 9.5e-29]], $MachinePrecision]], t$95$2, N[(1.0 / N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999927e-105 or 6.70000000000000027e-128 < z < 4e-116 or 9.50000000000000023e-29 < z

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

   3. Simplified 78.2%

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

      1. div-sub 78.2%

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

      2. associate-/l* 86.7%

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

   6. Applied egg-rr 86.7%

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

   7. Taylor expanded in t around 0 77.8%

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

      1. distribute-lft-out-- 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-/r* 79.0%

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

      4. div-sub 79.0%

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

      5. mul-1-neg 79.0%

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

   9. Simplified 79.0%

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

   if -9.99999999999999927e-105 < z < 6.70000000000000027e-128

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 80.2%

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

   if 4e-116 < z < 9.50000000000000023e-29

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. div-sub 99.6%

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

      2. associate-/l* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around inf 68.9%

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

      1. *-commutative 68.9%

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

   9. Simplified 68.9%

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

       1. clear-num 69.0%

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

       2. inv-pow 69.0%

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

   11. Applied egg-rr 69.0%

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

       1. unpow-1 69.0%

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

       2. *-commutative 69.0%

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

   13. Simplified 69.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-104}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-116} \lor \neg \left(z \leq 9.5 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t}{x - y \cdot z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-105} \lor \neg \left(z \leq 1.65 \cdot 10^{-128} \lor \neg \left(z \leq 4 \cdot 10^{-116}\right) \land z \leq 2.05 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -9.5e-105)
         (not (or (<= z 1.65e-128) (and (not (<= z 4e-116)) (<= z 2.05e-28)))))
   (/ (- y (/ x z)) a)
   (/ (- x (* y z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -9.5e-105) || !((z <= 1.65e-128) || (!(z <= 4e-116) && (z <= 2.05e-28)))) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * 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 ((z <= (-9.5d-105)) .or. (.not. (z <= 1.65d-128) .or. (.not. (z <= 4d-116)) .and. (z <= 2.05d-28))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (y * 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 ((z <= -9.5e-105) || !((z <= 1.65e-128) || (!(z <= 4e-116) && (z <= 2.05e-28)))) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -9.5e-105) or not ((z <= 1.65e-128) or (not (z <= 4e-116) and (z <= 2.05e-28))):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (y * z)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -9.5e-105) || !((z <= 1.65e-128) || (!(z <= 4e-116) && (z <= 2.05e-28))))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -9.5e-105) || ~(((z <= 1.65e-128) || (~((z <= 4e-116)) && (z <= 2.05e-28)))))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (y * z)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -9.5e-105], N[Not[Or[LessEqual[z, 1.65e-128], And[N[Not[LessEqual[z, 4e-116]], $MachinePrecision], LessEqual[z, 2.05e-28]]]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.5000000000000002e-105 or 1.65e-128 < z < 4e-116 or 2.0500000000000001e-28 < z

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

   3. Simplified 78.2%

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

      1. div-sub 78.2%

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

      2. associate-/l* 86.7%

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

   6. Applied egg-rr 86.7%

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

   7. Taylor expanded in t around 0 77.8%

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

      1. distribute-lft-out-- 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-/r* 79.0%

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

      4. div-sub 79.0%

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

      5. mul-1-neg 79.0%

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

   9. Simplified 79.0%

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

   if -9.5000000000000002e-105 < z < 1.65e-128 or 4e-116 < z < 2.0500000000000001e-28

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 78.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-105} \lor \neg \left(z \leq 1.65 \cdot 10^{-128} \lor \neg \left(z \leq 4 \cdot 10^{-116}\right) \land z \leq 2.05 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ \mathbf{if}\;z \leq 3.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{x - y \cdot z}{t\_1}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+198} \lor \neg \left(z \leq 10^{+267}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-z}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))))
   (if (<= z 3.7e+102)
     (/ (- x (* y z)) t_1)
     (if (or (<= z 1.75e+198) (not (<= z 1e+267)))
       (/ (- y (/ x z)) a)
       (* y (/ (- z) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double tmp;
	if (z <= 3.7e+102) {
		tmp = (x - (y * z)) / t_1;
	} else if ((z <= 1.75e+198) || !(z <= 1e+267)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = y * (-z / t_1);
	}
	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 = t - (z * a)
    if (z <= 3.7d+102) then
        tmp = (x - (y * z)) / t_1
    else if ((z <= 1.75d+198) .or. (.not. (z <= 1d+267))) then
        tmp = (y - (x / z)) / a
    else
        tmp = y * (-z / t_1)
    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 - (z * a);
	double tmp;
	if (z <= 3.7e+102) {
		tmp = (x - (y * z)) / t_1;
	} else if ((z <= 1.75e+198) || !(z <= 1e+267)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = y * (-z / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	tmp = 0
	if z <= 3.7e+102:
		tmp = (x - (y * z)) / t_1
	elif (z <= 1.75e+198) or not (z <= 1e+267):
		tmp = (y - (x / z)) / a
	else:
		tmp = y * (-z / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	tmp = 0.0
	if (z <= 3.7e+102)
		tmp = Float64(Float64(x - Float64(y * z)) / t_1);
	elseif ((z <= 1.75e+198) || !(z <= 1e+267))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(y * Float64(Float64(-z) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	tmp = 0.0;
	if (z <= 3.7e+102)
		tmp = (x - (y * z)) / t_1;
	elseif ((z <= 1.75e+198) || ~((z <= 1e+267)))
		tmp = (y - (x / z)) / a;
	else
		tmp = y * (-z / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3.7e+102], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[Or[LessEqual[z, 1.75e+198], N[Not[LessEqual[z, 1e+267]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(y * N[((-z) / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.70000000000000023e102

   1. Initial program 93.2%

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

   if 3.70000000000000023e102 < z < 1.75000000000000006e198 or 9.9999999999999997e266 < z

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

   3. Simplified 55.1%

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

      1. div-sub 55.1%

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

      2. associate-/l* 63.8%

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

   6. Applied egg-rr 63.8%

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

   7. Taylor expanded in t around 0 88.5%

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

      1. distribute-lft-out-- 88.5%

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

      2. *-commutative 88.5%

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

      3. associate-/r* 96.5%

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

      4. div-sub 96.5%

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

      5. mul-1-neg 96.5%

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

   9. Simplified 96.5%

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

   if 1.75000000000000006e198 < z < 9.9999999999999997e266

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

   3. Simplified 76.2%

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

      1. div-sub 76.2%

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

      2. associate-/l* 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. *-commutative 68.1%

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

      3. associate-*r/ 91.5%

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

      4. distribute-rgt-neg-in 91.5%

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

      5. distribute-neg-frac 91.5%

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

      6. *-commutative 91.5%

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

   9. Simplified 91.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+198} \lor \neg \left(z \leq 10^{+267}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-z}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+27} \lor \neg \left(z \leq 8.8 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -9.2e+27) (not (<= z 8.8e-29))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -9.2e+27) || !(z <= 8.8e-29)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * 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 <= (-9.2d+27)) .or. (.not. (z <= 8.8d-29))) then
        tmp = y / a
    else
        tmp = x / (t - (z * 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 <= -9.2e+27) || !(z <= 8.8e-29)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -9.2e+27) or not (z <= 8.8e-29):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -9.2e+27) || !(z <= 8.8e-29))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -9.2e+27) || ~((z <= 8.8e-29)))
		tmp = y / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -9.2e+27], N[Not[LessEqual[z, 8.8e-29]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.2000000000000002e27 or 8.79999999999999961e-29 < z

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in z around inf 63.4%

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

   if -9.2000000000000002e27 < z < 8.79999999999999961e-29

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 64.0%

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

      1. *-commutative 64.0%

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

   7. Simplified 64.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+27} \lor \neg \left(z \leq 8.8 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+28} \lor \neg \left(z \leq 8.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7e+28) (not (<= z 8.2e-29))) (/ y a) (/ (- x (* y z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7e+28) || !(z <= 8.2e-29)) {
		tmp = y / a;
	} else {
		tmp = (x - (y * 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 ((z <= (-7d+28)) .or. (.not. (z <= 8.2d-29))) then
        tmp = y / a
    else
        tmp = (x - (y * 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 ((z <= -7e+28) || !(z <= 8.2e-29)) {
		tmp = y / a;
	} else {
		tmp = (x - (y * z)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7e+28) or not (z <= 8.2e-29):
		tmp = y / a
	else:
		tmp = (x - (y * z)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7e+28) || !(z <= 8.2e-29))
		tmp = Float64(y / a);
	else
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -7e+28) || ~((z <= 8.2e-29)))
		tmp = y / a;
	else
		tmp = (x - (y * z)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7e+28], N[Not[LessEqual[z, 8.2e-29]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.9999999999999999e28 or 8.1999999999999996e-29 < z

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in z around inf 63.4%

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

   if -6.9999999999999999e28 < z < 8.1999999999999996e-29

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 69.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+28} \lor \neg \left(z \leq 8.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-105} \lor \neg \left(z \leq 6.7 \cdot 10^{-128}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7e-105) (not (<= z 6.7e-128))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7e-105) || !(z <= 6.7e-128)) {
		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 <= (-7d-105)) .or. (.not. (z <= 6.7d-128))) 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 <= -7e-105) || !(z <= 6.7e-128)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7e-105) or not (z <= 6.7e-128):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7e-105) || !(z <= 6.7e-128))
		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 <= -7e-105) || ~((z <= 6.7e-128)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7e-105], N[Not[LessEqual[z, 6.7e-128]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7e-105 or 6.70000000000000027e-128 < z

   1. Initial program 80.6%

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

      1. *-commutative 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in z around inf 56.2%

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

   if -7e-105 < z < 6.70000000000000027e-128

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 59.4%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-105} \lor \neg \left(z \leq 6.7 \cdot 10^{-128}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.7% accurate, 3.7× 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 87.3%

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

   1. *-commutative 87.3%

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

3. Simplified 87.3%

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

5. Taylor expanded in z around 0 29.5%

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

6. Final simplification 29.5%

   \[\leadsto \frac{x}{t} \]
7. Add Preprocessing

Developer target: 97.5% accurate, 0.4× 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 2024034 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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