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

Percentage Accurate: 85.3% → 97.3%

Time: 14.4s

Alternatives: 12

Speedup: 0.2×

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: 85.3% 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: 97.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - z \cdot a\\ t_3 := {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{t_2}}\right)}^{3} - y \cdot \frac{z}{t_2}\\ t_4 := \frac{t_1}{t_2}\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{+170}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_4 \leq 4 \cdot 10^{+219}:\\ \;\;\;\;{\left(\frac{t}{t_1} - \frac{a}{\frac{t_1}{z}}\right)}^{-1}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z)))
        (t_2 (- t (* z a)))
        (t_3 (- (pow (/ (cbrt x) (cbrt t_2)) 3.0) (* y (/ z t_2))))
        (t_4 (/ t_1 t_2)))
   (if (<= t_4 -2e+170)
     t_3
     (if (<= t_4 4e+219)
       (pow (- (/ t t_1) (/ a (/ t_1 z))) -1.0)
       (if (<= t_4 INFINITY) t_3 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = pow((cbrt(x) / cbrt(t_2)), 3.0) - (y * (z / t_2));
	double t_4 = t_1 / t_2;
	double tmp;
	if (t_4 <= -2e+170) {
		tmp = t_3;
	} else if (t_4 <= 4e+219) {
		tmp = pow(((t / t_1) - (a / (t_1 / z))), -1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = Math.pow((Math.cbrt(x) / Math.cbrt(t_2)), 3.0) - (y * (z / t_2));
	double t_4 = t_1 / t_2;
	double tmp;
	if (t_4 <= -2e+170) {
		tmp = t_3;
	} else if (t_4 <= 4e+219) {
		tmp = Math.pow(((t / t_1) - (a / (t_1 / z))), -1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64((Float64(cbrt(x) / cbrt(t_2)) ^ 3.0) - Float64(y * Float64(z / t_2)))
	t_4 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_4 <= -2e+170)
		tmp = t_3;
	elseif (t_4 <= 4e+219)
		tmp = Float64(Float64(t / t_1) - Float64(a / Float64(t_1 / z))) ^ -1.0;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(y / a);
	end
	return 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[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[Power[x, 1/3], $MachinePrecision] / N[Power[t$95$2, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] - N[(y * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+170], t$95$3, If[LessEqual[t$95$4, 4e+219], N[Power[N[(N[(t / t$95$1), $MachinePrecision] - N[(a / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -2.00000000000000007e170 or 3.99999999999999986e219 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

   3. Simplified 74.9%

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

      1. div-sub 74.9%

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

      2. sub-neg 74.9%

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

      3. add-cube-cbrt 74.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{t - z \cdot a} + \left(-\frac{y \cdot z}{t - z \cdot a}\right) \]

      4. add-cube-cbrt 74.3%

         \[\leadsto \frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}{\color{blue}{\left(\sqrt[3]{t - z \cdot a} \cdot \sqrt[3]{t - z \cdot a}\right) \cdot \sqrt[3]{t - z \cdot a}}} + \left(-\frac{y \cdot z}{t - z \cdot a}\right) \]

      5. times-frac 74.3%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{t - z \cdot a} \cdot \sqrt[3]{t - z \cdot a}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t - z \cdot a}}} + \left(-\frac{y \cdot z}{t - z \cdot a}\right) \]

      6. fma-def 74.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{t - z \cdot a} \cdot \sqrt[3]{t - z \cdot a}}, \frac{\sqrt[3]{x}}{\sqrt[3]{t - z \cdot a}}, -\frac{y \cdot z}{t - z \cdot a}\right)} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{{\left(\sqrt[3]{\mathsf{fma}\left(a, -z, t\right)}\right)}^{2}}, \frac{\sqrt[3]{x}}{\sqrt[3]{\mathsf{fma}\left(a, -z, t\right)}}, -\frac{y}{\frac{\mathsf{fma}\left(a, -z, t\right)}{z}}\right)} \]
   6. Step-by-step derivation

      1. fma-neg 99.2%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{{\left(\sqrt[3]{\mathsf{fma}\left(a, -z, t\right)}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\mathsf{fma}\left(a, -z, t\right)}} - \frac{y}{\frac{\mathsf{fma}\left(a, -z, t\right)}{z}}} \]

      2. *-commutative 99.2%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{\sqrt[3]{\mathsf{fma}\left(a, -z, t\right)}} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{2}}{{\left(\sqrt[3]{\mathsf{fma}\left(a, -z, t\right)}\right)}^{2}}} - \frac{y}{\frac{\mathsf{fma}\left(a, -z, t\right)}{z}} \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{t - z \cdot a}}\right)}^{3} - y \cdot \frac{z}{t - z \cdot a}} \]

   if -2.00000000000000007e170 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 3.99999999999999986e219

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

   3. Simplified 89.3%

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

      1. clear-num 89.2%

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

      2. inv-pow 89.2%

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

      3. sub-neg 89.2%

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

      4. +-commutative 89.2%

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

      5. *-commutative 89.2%

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

      6. distribute-rgt-neg-in 89.2%

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

      7. fma-def 89.2%

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

   5. Applied egg-rr 89.2%

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

   6. Taylor expanded in a around 0 89.2%

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

      1. +-commutative 89.2%

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

      2. mul-1-neg 89.2%

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

      3. unsub-neg 89.2%

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

      4. associate-/l* 99.6%

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

   8. Simplified 99.6%

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

   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. Taylor expanded in z around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{+170}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{t - z \cdot a}}\right)}^{3} - y \cdot \frac{z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 4 \cdot 10^{+219}:\\ \;\;\;\;{\left(\frac{t}{x - y \cdot z} - \frac{a}{\frac{x - y \cdot z}{z}}\right)}^{-1}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{t - z \cdot a}}\right)}^{3} - y \cdot \frac{z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{-y}{\frac{t_1}{z}}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{y \cdot \frac{t}{{a}^{2}} - \frac{x}{a}}{z}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 (- INFINITY))
     (/ (- y) (/ t_1 z))
     (if (<= t_2 -2e-304)
       t_2
       (if (<= t_2 0.0)
         (+ (/ y a) (/ (- (* y (/ t (pow a 2.0))) (/ x a)) z))
         (if (<= t_2 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 - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -y / (t_1 / z);
	} else if (t_2 <= -2e-304) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (y / a) + (((y * (t / pow(a, 2.0))) - (x / a)) / z);
	} else if (t_2 <= ((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 - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -y / (t_1 / z);
	} else if (t_2 <= -2e-304) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (y / a) + (((y * (t / Math.pow(a, 2.0))) - (x / a)) / z);
	} else if (t_2 <= 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 - (y * z)) / t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -y / (t_1 / z)
	elif t_2 <= -2e-304:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = (y / a) + (((y * (t / math.pow(a, 2.0))) - (x / a)) / z)
	elif t_2 <= 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 - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-y) / Float64(t_1 / z));
	elseif (t_2 <= -2e-304)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(y / a) + Float64(Float64(Float64(y * Float64(t / (a ^ 2.0))) - Float64(x / a)) / z));
	elseif (t_2 <= 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 - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -y / (t_1 / z);
	elseif (t_2 <= -2e-304)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = (y / a) + (((y * (t / (a ^ 2.0))) - (x / a)) / z);
	elseif (t_2 <= 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 - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[((-y) / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-304], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(y / a), $MachinePrecision] + N[(N[(N[(y * N[(t / N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 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

   1. Initial program 46.3%

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

      1. *-commutative 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in x around 0 23.2%

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

      1. mul-1-neg 23.2%

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

      2. associate-/l* 76.8%

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

      3. *-commutative 76.8%

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

   6. Simplified 76.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.99999999999999994e-304 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 97.7%

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

   if -1.99999999999999994e-304 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 41.4%

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

      1. *-commutative 41.4%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in z around inf 51.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{a \cdot z} + \frac{y}{a}\right) - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}} \]
   5. Step-by-step derivation

      1. +-commutative 51.3%

         \[\leadsto \color{blue}{\left(\frac{y}{a} + -1 \cdot \frac{x}{a \cdot z}\right)} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z} \]

      2. associate--l+ 51.3%

         \[\leadsto \color{blue}{\frac{y}{a} + \left(-1 \cdot \frac{x}{a \cdot z} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right)} \]

      3. associate-/r* 87.8%

         \[\leadsto \frac{y}{a} + \left(-1 \cdot \color{blue}{\frac{\frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      4. associate-*r/ 87.8%

         \[\leadsto \frac{y}{a} + \left(\color{blue}{\frac{-1 \cdot \frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      5. associate-/r* 87.8%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - -1 \cdot \color{blue}{\frac{\frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      6. associate-*r/ 87.8%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - \color{blue}{\frac{-1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      7. div-sub 87.8%

         \[\leadsto \frac{y}{a} + \color{blue}{\frac{-1 \cdot \frac{x}{a} - -1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}} \]

      8. distribute-lft-out-- 87.8%

         \[\leadsto \frac{y}{a} + \frac{\color{blue}{-1 \cdot \left(\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}\right)}}{z} \]

      9. associate-*r/ 87.8%

         \[\leadsto \frac{y}{a} + \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      10. mul-1-neg 87.8%

          \[\leadsto \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)} \]

      11. unsub-neg 87.8%

          \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

   6. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t}{{a}^{2}} \cdot y}{z}} \]

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

   1. Initial program 0.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. 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. Taylor expanded in z around inf 100.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{-y}{\frac{t - z \cdot a}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{y \cdot \frac{t}{{a}^{2}} - \frac{x}{a}}{z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 3: 92.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - z \cdot a\\ t_3 := \frac{t_1}{t_2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{-y}{\frac{t_2}{z}}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;{\left(\frac{t}{t_1} - \frac{a}{\frac{t_1}{z}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (- t (* z a))) (t_3 (/ t_1 t_2)))
   (if (<= t_3 (- INFINITY))
     (/ (- y) (/ t_2 z))
     (if (<= t_3 INFINITY) (pow (- (/ t t_1) (/ a (/ t_1 z))) -1.0) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -y / (t_2 / z);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = pow(((t / t_1) - (a / (t_1 / z))), -1.0);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -y / (t_2 / z);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow(((t / t_1) - (a / (t_1 / z))), -1.0);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - (y * z);
	t_2 = t - (z * a);
	t_3 = t_1 / t_2;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -y / (t_2 / z);
	elseif (t_3 <= Inf)
		tmp = ((t / t_1) - (a / (t_1 / z))) ^ -1.0;
	else
		tmp = y / a;
	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[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[((-y) / N[(t$95$2 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Power[N[(N[(t / t$95$1), $MachinePrecision] - N[(a / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 46.3%

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

      1. *-commutative 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in x around 0 23.2%

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

      1. mul-1-neg 23.2%

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

      2. associate-/l* 76.8%

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

      3. *-commutative 76.8%

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

   6. Simplified 76.8%

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

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

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

   3. Simplified 89.5%

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

      1. clear-num 89.3%

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

      2. inv-pow 89.3%

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

      3. sub-neg 89.3%

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

      4. +-commutative 89.3%

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

      5. *-commutative 89.3%

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

      6. distribute-rgt-neg-in 89.3%

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

      7. fma-def 89.3%

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

   5. Applied egg-rr 89.3%

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

   6. Taylor expanded in a around 0 89.3%

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

      1. +-commutative 89.3%

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

      2. mul-1-neg 89.3%

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

      3. unsub-neg 89.3%

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

      4. associate-/l* 96.6%

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

   8. Simplified 96.6%

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

   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. Taylor expanded in z around inf 100.0%

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

4. Final simplification 95.2%

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

Alternative 4: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{-y}{\frac{t_1}{z}}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 (- INFINITY))
     (/ (- y) (/ t_1 z))
     (if (<= t_2 -2e-304)
       t_2
       (if (<= t_2 0.0)
         (/ (- y (/ x z)) a)
         (if (<= t_2 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 - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -y / (t_1 / z);
	} else if (t_2 <= -2e-304) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_2 <= ((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 - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -y / (t_1 / z);
	} else if (t_2 <= -2e-304) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_2 <= 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 - (y * z)) / t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -y / (t_1 / z)
	elif t_2 <= -2e-304:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = (y - (x / z)) / a
	elif t_2 <= 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 - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-y) / Float64(t_1 / z));
	elseif (t_2 <= -2e-304)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (t_2 <= 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 - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -y / (t_1 / z);
	elseif (t_2 <= -2e-304)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = (y - (x / z)) / a;
	elseif (t_2 <= 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 - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[((-y) / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-304], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$2, 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

   1. Initial program 46.3%

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

      1. *-commutative 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in x around 0 23.2%

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

      1. mul-1-neg 23.2%

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

      2. associate-/l* 76.8%

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

      3. *-commutative 76.8%

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

   6. Simplified 76.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.99999999999999994e-304 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 97.7%

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

   if -1.99999999999999994e-304 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 41.4%

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

      1. *-commutative 41.4%

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

   3. Simplified 41.4%

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

      1. clear-num 41.4%

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

      2. associate-/r/ 41.4%

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

      3. sub-neg 41.4%

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

      4. +-commutative 41.4%

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

      5. *-commutative 41.4%

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

      6. distribute-rgt-neg-in 41.4%

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

      7. fma-def 41.4%

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

   5. Applied egg-rr 41.4%

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

   6. Taylor expanded in z around -inf 87.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z} + \frac{y}{a}} \]
   7. Step-by-step derivation

      1. +-commutative 87.8%

         \[\leadsto \color{blue}{\frac{y}{a} + -1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      2. mul-1-neg 87.8%

         \[\leadsto \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)} \]

      3. unsub-neg 87.8%

         \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      4. associate-/l* 90.9%

         \[\leadsto \frac{y}{a} - \frac{\frac{x}{a} - \color{blue}{\frac{t}{\frac{{a}^{2}}{y}}}}{z} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t}{\frac{{a}^{2}}{y}}}{z}} \]

   9. Taylor expanded in a around inf 88.0%

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

   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. Taylor expanded in z around inf 100.0%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{-y}{\frac{t - z \cdot a}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5: 72.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := t - z \cdot a\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 246000:\\ \;\;\;\;\frac{x}{t_2}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+40}:\\ \;\;\;\;\frac{-y}{\frac{t_2}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)) (t_2 (- t (* z a))))
   (if (<= z -1.6e-53)
     t_1
     (if (<= z 2.1e-124)
       (/ (- x (* y z)) t)
       (if (<= z 246000.0)
         (/ x t_2)
         (if (<= z 1.1e+40) (/ (- y) (/ t_2 z)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double t_2 = t - (z * a);
	double tmp;
	if (z <= -1.6e-53) {
		tmp = t_1;
	} else if (z <= 2.1e-124) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 246000.0) {
		tmp = x / t_2;
	} else if (z <= 1.1e+40) {
		tmp = -y / (t_2 / z);
	} else {
		tmp = 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 = (y - (x / z)) / a
    t_2 = t - (z * a)
    if (z <= (-1.6d-53)) then
        tmp = t_1
    else if (z <= 2.1d-124) then
        tmp = (x - (y * z)) / t
    else if (z <= 246000.0d0) then
        tmp = x / t_2
    else if (z <= 1.1d+40) then
        tmp = -y / (t_2 / z)
    else
        tmp = 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 = (y - (x / z)) / a;
	double t_2 = t - (z * a);
	double tmp;
	if (z <= -1.6e-53) {
		tmp = t_1;
	} else if (z <= 2.1e-124) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 246000.0) {
		tmp = x / t_2;
	} else if (z <= 1.1e+40) {
		tmp = -y / (t_2 / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	t_2 = t - (z * a)
	tmp = 0
	if z <= -1.6e-53:
		tmp = t_1
	elif z <= 2.1e-124:
		tmp = (x - (y * z)) / t
	elif z <= 246000.0:
		tmp = x / t_2
	elif z <= 1.1e+40:
		tmp = -y / (t_2 / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	t_2 = Float64(t - Float64(z * a))
	tmp = 0.0
	if (z <= -1.6e-53)
		tmp = t_1;
	elseif (z <= 2.1e-124)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 246000.0)
		tmp = Float64(x / t_2);
	elseif (z <= 1.1e+40)
		tmp = Float64(Float64(-y) / Float64(t_2 / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	t_2 = t - (z * a);
	tmp = 0.0;
	if (z <= -1.6e-53)
		tmp = t_1;
	elseif (z <= 2.1e-124)
		tmp = (x - (y * z)) / t;
	elseif (z <= 246000.0)
		tmp = x / t_2;
	elseif (z <= 1.1e+40)
		tmp = -y / (t_2 / z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.6e-53], t$95$1, If[LessEqual[z, 2.1e-124], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 246000.0], N[(x / t$95$2), $MachinePrecision], If[LessEqual[z, 1.1e+40], N[((-y) / N[(t$95$2 / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.6e-53 or 1.0999999999999999e40 < z

   1. Initial program 66.4%

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

      1. *-commutative 66.4%

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

   3. Simplified 66.4%

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

      1. clear-num 66.3%

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

      2. associate-/r/ 66.3%

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

      3. sub-neg 66.3%

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

      4. +-commutative 66.3%

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

      5. *-commutative 66.3%

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

      6. distribute-rgt-neg-in 66.3%

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

      7. fma-def 66.3%

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

   5. Applied egg-rr 66.3%

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

   6. Taylor expanded in z around -inf 60.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z} + \frac{y}{a}} \]
   7. Step-by-step derivation

      1. +-commutative 60.6%

         \[\leadsto \color{blue}{\frac{y}{a} + -1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      2. mul-1-neg 60.6%

         \[\leadsto \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)} \]

      3. unsub-neg 60.6%

         \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      4. associate-/l* 65.4%

         \[\leadsto \frac{y}{a} - \frac{\frac{x}{a} - \color{blue}{\frac{t}{\frac{{a}^{2}}{y}}}}{z} \]

   8. Simplified 65.4%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t}{\frac{{a}^{2}}{y}}}{z}} \]

   9. Taylor expanded in a around inf 75.9%

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

   if -1.6e-53 < z < 2.1000000000000001e-124

   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. Taylor expanded in t around inf 84.2%

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

   if 2.1000000000000001e-124 < z < 246000

   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. Taylor expanded in x around inf 69.9%

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

      1. *-commutative 69.9%

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

   6. Simplified 69.9%

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

   if 246000 < z < 1.0999999999999999e40

   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. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. associate-/l* 99.5%

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

      3. *-commutative 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 246000:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+40}:\\ \;\;\;\;\frac{-y}{\frac{t - z \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 6: 59.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+183} \lor \neg \left(y \leq -1.18 \cdot 10^{+141} \lor \neg \left(y \leq -9.6 \cdot 10^{-5}\right) \land y \leq 9.6 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -3.5e+183)
         (not
          (or (<= y -1.18e+141) (and (not (<= y -9.6e-5)) (<= y 9.6e+41)))))
   (/ y a)
   (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -3.5e+183) || !((y <= -1.18e+141) || (!(y <= -9.6e-5) && (y <= 9.6e+41)))) {
		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 ((y <= (-3.5d+183)) .or. (.not. (y <= (-1.18d+141)) .or. (.not. (y <= (-9.6d-5))) .and. (y <= 9.6d+41))) 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 ((y <= -3.5e+183) || !((y <= -1.18e+141) || (!(y <= -9.6e-5) && (y <= 9.6e+41)))) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -3.5e+183) or not ((y <= -1.18e+141) or (not (y <= -9.6e-5) and (y <= 9.6e+41))):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -3.5e+183) || !((y <= -1.18e+141) || (!(y <= -9.6e-5) && (y <= 9.6e+41))))
		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 ((y <= -3.5e+183) || ~(((y <= -1.18e+141) || (~((y <= -9.6e-5)) && (y <= 9.6e+41)))))
		tmp = y / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -3.5e+183], N[Not[Or[LessEqual[y, -1.18e+141], And[N[Not[LessEqual[y, -9.6e-5]], $MachinePrecision], LessEqual[y, 9.6e+41]]]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999987e183 or -1.1800000000000001e141 < y < -9.6000000000000002e-5 or 9.6000000000000007e41 < y

   1. Initial program 69.4%

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

      1. *-commutative 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in z around inf 62.0%

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

   if -3.49999999999999987e183 < y < -1.1800000000000001e141 or -9.6000000000000002e-5 < y < 9.6000000000000007e41

   1. Initial program 88.2%

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

      1. *-commutative 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. *-commutative 69.2%

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

   6. Simplified 69.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+183} \lor \neg \left(y \leq -1.18 \cdot 10^{+141} \lor \neg \left(y \leq -9.6 \cdot 10^{-5}\right) \land y \leq 9.6 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 7: 59.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+227}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq -4 \cdot 10^{+140}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-8} \lor \neg \left(y \leq 2.05 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -2.2e+227)
   (/ y a)
   (if (<= y -4e+140)
     (/ (- x (* y z)) t)
     (if (or (<= y -2.05e-8) (not (<= y 2.05e+41)))
       (/ y a)
       (/ x (- t (* z a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.2e+227) {
		tmp = y / a;
	} else if (y <= -4e+140) {
		tmp = (x - (y * z)) / t;
	} else if ((y <= -2.05e-8) || !(y <= 2.05e+41)) {
		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 (y <= (-2.2d+227)) then
        tmp = y / a
    else if (y <= (-4d+140)) then
        tmp = (x - (y * z)) / t
    else if ((y <= (-2.05d-8)) .or. (.not. (y <= 2.05d+41))) 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 (y <= -2.2e+227) {
		tmp = y / a;
	} else if (y <= -4e+140) {
		tmp = (x - (y * z)) / t;
	} else if ((y <= -2.05e-8) || !(y <= 2.05e+41)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -2.2e+227:
		tmp = y / a
	elif y <= -4e+140:
		tmp = (x - (y * z)) / t
	elif (y <= -2.05e-8) or not (y <= 2.05e+41):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -2.2e+227)
		tmp = Float64(y / a);
	elseif (y <= -4e+140)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif ((y <= -2.05e-8) || !(y <= 2.05e+41))
		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 (y <= -2.2e+227)
		tmp = y / a;
	elseif (y <= -4e+140)
		tmp = (x - (y * z)) / t;
	elseif ((y <= -2.05e-8) || ~((y <= 2.05e+41)))
		tmp = y / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -2.2e+227], N[(y / a), $MachinePrecision], If[LessEqual[y, -4e+140], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[Or[LessEqual[y, -2.05e-8], N[Not[LessEqual[y, 2.05e+41]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2000000000000002e227 or -4.00000000000000024e140 < y < -2.05000000000000016e-8 or 2.0500000000000002e41 < y

   1. Initial program 67.9%

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

      1. *-commutative 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in z around inf 62.9%

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

   if -2.2000000000000002e227 < y < -4.00000000000000024e140

   1. Initial program 84.8%

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

      1. *-commutative 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in t around inf 59.1%

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

   if -2.05000000000000016e-8 < y < 2.0500000000000002e41

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around inf 69.5%

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

      1. *-commutative 69.5%

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

   6. Simplified 69.5%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+227}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq -4 \cdot 10^{+140}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-8} \lor \neg \left(y \leq 2.05 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 8: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 14500:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6e+20)
   (/ y a)
   (if (<= z -2.2e-54)
     (/ (- x) (* z a))
     (if (<= z 14500.0)
       (/ x t)
       (if (<= z 2.7e+136) (/ (/ (- x) a) z) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+20) {
		tmp = y / a;
	} else if (z <= -2.2e-54) {
		tmp = -x / (z * a);
	} else if (z <= 14500.0) {
		tmp = x / t;
	} else if (z <= 2.7e+136) {
		tmp = (-x / a) / z;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6d+20)) then
        tmp = y / a
    else if (z <= (-2.2d-54)) then
        tmp = -x / (z * a)
    else if (z <= 14500.0d0) then
        tmp = x / t
    else if (z <= 2.7d+136) then
        tmp = (-x / a) / z
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+20) {
		tmp = y / a;
	} else if (z <= -2.2e-54) {
		tmp = -x / (z * a);
	} else if (z <= 14500.0) {
		tmp = x / t;
	} else if (z <= 2.7e+136) {
		tmp = (-x / a) / z;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6e+20:
		tmp = y / a
	elif z <= -2.2e-54:
		tmp = -x / (z * a)
	elif z <= 14500.0:
		tmp = x / t
	elif z <= 2.7e+136:
		tmp = (-x / a) / z
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6e+20)
		tmp = Float64(y / a);
	elseif (z <= -2.2e-54)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 14500.0)
		tmp = Float64(x / t);
	elseif (z <= 2.7e+136)
		tmp = Float64(Float64(Float64(-x) / a) / 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 <= -6e+20)
		tmp = y / a;
	elseif (z <= -2.2e-54)
		tmp = -x / (z * a);
	elseif (z <= 14500.0)
		tmp = x / t;
	elseif (z <= 2.7e+136)
		tmp = (-x / a) / z;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6e+20], N[(y / a), $MachinePrecision], If[LessEqual[z, -2.2e-54], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 14500.0], N[(x / t), $MachinePrecision], If[LessEqual[z, 2.7e+136], N[(N[((-x) / a), $MachinePrecision] / z), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6e20 or 2.7000000000000002e136 < z

   1. Initial program 58.8%

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

      1. *-commutative 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in z around inf 60.5%

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

   if -6e20 < z < -2.2e-54

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a around inf 82.3%

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

      1. associate-/r* 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in z around 0 66.4%

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

       1. mul-1-neg 66.4%

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

       2. *-commutative 66.4%

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

   11. Simplified 66.4%

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

   if -2.2e-54 < z < 14500

   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. Taylor expanded in z around 0 61.1%

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

   if 14500 < z < 2.7000000000000002e136

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

   3. Simplified 85.2%

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

      1. clear-num 85.2%

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

      2. associate-/r/ 85.2%

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

      3. sub-neg 85.2%

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

      4. +-commutative 85.2%

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

      5. *-commutative 85.2%

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

      6. distribute-rgt-neg-in 85.2%

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

      7. fma-def 85.2%

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

   5. Applied egg-rr 85.2%

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

   6. Taylor expanded in a around inf 55.9%

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

      1. associate-/r* 58.6%

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

   8. Simplified 58.6%

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

   9. Taylor expanded in z around 0 36.8%

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

       1. mul-1-neg 36.8%

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

       2. associate-/r* 51.1%

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

       3. distribute-neg-frac 51.1%

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

       4. distribute-neg-frac 51.1%

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

   11. Simplified 51.1%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 14500:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-58} \lor \neg \left(z \leq 3.1 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -8e-58) (not (<= z 3.1e-10)))
   (/ (- 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 <= -8e-58) || !(z <= 3.1e-10)) {
		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 <= (-8d-58)) .or. (.not. (z <= 3.1d-10))) 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 <= -8e-58) || !(z <= 3.1e-10)) {
		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 <= -8e-58) or not (z <= 3.1e-10):
		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 <= -8e-58) || !(z <= 3.1e-10))
		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 <= -8e-58) || ~((z <= 3.1e-10)))
		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, -8e-58], N[Not[LessEqual[z, 3.1e-10]], $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 < -8.0000000000000002e-58 or 3.10000000000000015e-10 < z

   1. Initial program 67.7%

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

      1. *-commutative 67.7%

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

   3. Simplified 67.7%

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

      1. clear-num 67.6%

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

      2. associate-/r/ 67.6%

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

      3. sub-neg 67.6%

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

      4. +-commutative 67.6%

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

      5. *-commutative 67.6%

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

      6. distribute-rgt-neg-in 67.6%

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

      7. fma-def 67.6%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in z around -inf 60.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z} + \frac{y}{a}} \]
   7. Step-by-step derivation

      1. +-commutative 60.2%

         \[\leadsto \color{blue}{\frac{y}{a} + -1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      2. mul-1-neg 60.2%

         \[\leadsto \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)} \]

      3. unsub-neg 60.2%

         \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      4. associate-/l* 65.5%

         \[\leadsto \frac{y}{a} - \frac{\frac{x}{a} - \color{blue}{\frac{t}{\frac{{a}^{2}}{y}}}}{z} \]

   8. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t}{\frac{{a}^{2}}{y}}}{z}} \]

   9. Taylor expanded in a around inf 75.6%

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

   if -8.0000000000000002e-58 < z < 3.10000000000000015e-10

   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. Taylor expanded in t around inf 77.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-58} \lor \neg \left(z \leq 3.1 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]

Alternative 10: 55.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.25e+21)
   (/ y a)
   (if (<= z -5.2e-58) (/ (- x) (* z a)) (if (<= z 6.6e-8) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e+21) {
		tmp = y / a;
	} else if (z <= -5.2e-58) {
		tmp = -x / (z * a);
	} else if (z <= 6.6e-8) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.25d+21)) then
        tmp = y / a
    else if (z <= (-5.2d-58)) then
        tmp = -x / (z * a)
    else if (z <= 6.6d-8) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e+21) {
		tmp = y / a;
	} else if (z <= -5.2e-58) {
		tmp = -x / (z * a);
	} else if (z <= 6.6e-8) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.25e+21:
		tmp = y / a
	elif z <= -5.2e-58:
		tmp = -x / (z * a)
	elif z <= 6.6e-8:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.25e+21)
		tmp = Float64(y / a);
	elseif (z <= -5.2e-58)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 6.6e-8)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.25e+21)
		tmp = y / a;
	elseif (z <= -5.2e-58)
		tmp = -x / (z * a);
	elseif (z <= 6.6e-8)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.25e+21], N[(y / a), $MachinePrecision], If[LessEqual[z, -5.2e-58], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.6e-8], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25e21 or 6.59999999999999954e-8 < z

   1. Initial program 63.1%

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

      1. *-commutative 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in z around inf 55.1%

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

   if -1.25e21 < z < -5.20000000000000013e-58

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a around inf 82.3%

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

      1. associate-/r* 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in z around 0 66.4%

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

       1. mul-1-neg 66.4%

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

       2. *-commutative 66.4%

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

   11. Simplified 66.4%

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

   if -5.20000000000000013e-58 < z < 6.59999999999999954e-8

   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. Taylor expanded in z around 0 61.6%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 11: 55.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-15} \lor \neg \left(z \leq 0.212\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.2e-15) (not (<= z 0.212))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.2e-15) || !(z <= 0.212)) {
		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 <= (-4.2d-15)) .or. (.not. (z <= 0.212d0))) 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 <= -4.2e-15) || !(z <= 0.212)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.2e-15) or not (z <= 0.212):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.2e-15) || !(z <= 0.212))
		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 <= -4.2e-15) || ~((z <= 0.212)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.2e-15], N[Not[LessEqual[z, 0.212]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999962e-15 or 0.211999999999999994 < z

   1. Initial program 65.7%

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

      1. *-commutative 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in z around inf 55.0%

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

   if -4.19999999999999962e-15 < z < 0.211999999999999994

   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. Taylor expanded in z around 0 59.2%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-15} \lor \neg \left(z \leq 0.212\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 12: 35.6% 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 80.7%

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

   1. *-commutative 80.7%

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

3. Simplified 80.7%

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

4. Taylor expanded in z around 0 32.7%

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

5. Final simplification 32.7%

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

Developer target: 97.2% accurate, 0.6× 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 2023331 
(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))))