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

Percentage Accurate: 84.3% → 98.4%

Time: 13.9s

Alternatives: 12

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.3% 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: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - z \cdot a\\ t_3 := \mathsf{fma}\left(-1, \frac{y}{\frac{t_2}{z}}, \frac{x}{t_2}\right)\\ t_4 := \frac{t_1}{t_2}\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{-94}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_4 \leq 10^{-165}:\\ \;\;\;\;{\left(a \cdot \left(z \cdot \frac{-1}{t_1}\right) + \frac{t}{t_1}\right)}^{-1}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z)))
        (t_2 (- t (* z a)))
        (t_3 (fma -1.0 (/ y (/ t_2 z)) (/ x t_2)))
        (t_4 (/ t_1 t_2)))
   (if (<= t_4 -2e-94)
     t_3
     (if (<= t_4 1e-165)
       (pow (+ (* a (* z (/ -1.0 t_1))) (/ t t_1)) -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 = fma(-1.0, (y / (t_2 / z)), (x / t_2));
	double t_4 = t_1 / t_2;
	double tmp;
	if (t_4 <= -2e-94) {
		tmp = t_3;
	} else if (t_4 <= 1e-165) {
		tmp = pow(((a * (z * (-1.0 / t_1))) + (t / t_1)), -1.0);
	} else if (t_4 <= ((double) 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 = fma(-1.0, Float64(y / Float64(t_2 / z)), Float64(x / t_2))
	t_4 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_4 <= -2e-94)
		tmp = t_3;
	elseif (t_4 <= 1e-165)
		tmp = Float64(Float64(a * Float64(z * Float64(-1.0 / t_1))) + Float64(t / t_1)) ^ -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[(-1.0 * N[(y / N[(t$95$2 / z), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, -2e-94], t$95$3, If[LessEqual[t$95$4, 1e-165], N[Power[N[(N[(a * N[(z * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / t$95$1), $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))) < -1.9999999999999999e-94 or 1e-165 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 92.9%

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

      1. *-commutative 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in x around 0 92.8%

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

      1. fma-def 92.8%

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

      2. associate-/l* 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if -1.9999999999999999e-94 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1e-165

   1. Initial program 78.7%

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

      1. *-commutative 78.7%

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

   3. Simplified 78.7%

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

      1. clear-num 77.2%

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

      2. inv-pow 77.2%

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

      3. sub-neg 77.2%

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

      4. +-commutative 77.2%

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

      5. *-commutative 77.2%

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

      6. distribute-rgt-neg-in 77.2%

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

      7. fma-def 77.2%

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

   5. Applied egg-rr 77.2%

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

   6. Taylor expanded in a around 0 77.2%

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

      1. div-inv 77.3%

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

      2. associate-*l* 97.9%

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

      3. *-commutative 97.9%

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

   8. Applied egg-rr 97.9%

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

   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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{y}{\frac{t - z \cdot a}{z}}, \frac{x}{t - z \cdot a}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{-165}:\\ \;\;\;\;{\left(a \cdot \left(z \cdot \frac{-1}{x - y \cdot z}\right) + \frac{t}{x - y \cdot z}\right)}^{-1}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{y}{\frac{t - z \cdot a}{z}}, \frac{x}{t - z \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2: 94.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \mathsf{fma}\left(-1, \frac{y}{\frac{t_1}{z}}, \frac{x}{t_1}\right)\\ t_3 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{-304}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a)))
        (t_2 (fma -1.0 (/ y (/ t_1 z)) (/ x t_1)))
        (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 -5e-304)
     t_2
     (if (<= t_3 0.0)
       (- (/ y a) (/ (/ x z) a))
       (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 = fma(-1.0, (y / (t_1 / z)), (x / t_1));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -5e-304) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = (y / a) - ((x / z) / a);
	} 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 = fma(-1.0, Float64(y / Float64(t_1 / z)), Float64(x / t_1))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= -5e-304)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / z) / a));
	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[(-1.0 * N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-304], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(y / a), $MachinePrecision] - N[(N[(x / z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99999999999999965e-304 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 93.9%

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

      1. *-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in x around 0 93.9%

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

      1. fma-def 93.9%

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

      2. associate-/l* 97.9%

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

      3. *-commutative 97.9%

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

      4. *-commutative 97.9%

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

   6. Simplified 97.9%

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

   if -4.99999999999999965e-304 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 47.9%

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

      1. *-commutative 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in t around 0 39.8%

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

      1. associate-*r* 39.8%

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

      2. neg-mul-1 39.8%

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

      3. *-commutative 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in x around 0 72.5%

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

      1. +-commutative 72.5%

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

      2. mul-1-neg 72.5%

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

      3. unsub-neg 72.5%

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

      4. *-commutative 72.5%

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

      5. associate-/r* 91.8%

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

   9. Simplified 91.8%

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

   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 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{y}{\frac{t - z \cdot a}{z}}, \frac{x}{t - z \cdot a}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{y}{\frac{t - z \cdot a}{z}}, \frac{x}{t - z \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 3: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-39}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))) (t_2 (/ y (- a (/ t z)))))
   (if (<= z -4.4e+74)
     t_2
     (if (<= z -8.2e-74)
       t_1
       (if (<= z 1e-39)
         (/ (- x (* y z)) t)
         (if (<= z 1.55e+25)
           t_1
           (if (<= z 3.4e+101) t_2 (- (/ y a) (/ (/ x z) a)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double t_2 = y / (a - (t / z));
	double tmp;
	if (z <= -4.4e+74) {
		tmp = t_2;
	} else if (z <= -8.2e-74) {
		tmp = t_1;
	} else if (z <= 1e-39) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 1.55e+25) {
		tmp = t_1;
	} else if (z <= 3.4e+101) {
		tmp = t_2;
	} else {
		tmp = (y / a) - ((x / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / (t - (z * a))
    t_2 = y / (a - (t / z))
    if (z <= (-4.4d+74)) then
        tmp = t_2
    else if (z <= (-8.2d-74)) then
        tmp = t_1
    else if (z <= 1d-39) then
        tmp = (x - (y * z)) / t
    else if (z <= 1.55d+25) then
        tmp = t_1
    else if (z <= 3.4d+101) then
        tmp = t_2
    else
        tmp = (y / a) - ((x / z) / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double t_2 = y / (a - (t / z));
	double tmp;
	if (z <= -4.4e+74) {
		tmp = t_2;
	} else if (z <= -8.2e-74) {
		tmp = t_1;
	} else if (z <= 1e-39) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 1.55e+25) {
		tmp = t_1;
	} else if (z <= 3.4e+101) {
		tmp = t_2;
	} else {
		tmp = (y / a) - ((x / z) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	t_2 = y / (a - (t / z))
	tmp = 0
	if z <= -4.4e+74:
		tmp = t_2
	elif z <= -8.2e-74:
		tmp = t_1
	elif z <= 1e-39:
		tmp = (x - (y * z)) / t
	elif z <= 1.55e+25:
		tmp = t_1
	elif z <= 3.4e+101:
		tmp = t_2
	else:
		tmp = (y / a) - ((x / z) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	t_2 = Float64(y / Float64(a - Float64(t / z)))
	tmp = 0.0
	if (z <= -4.4e+74)
		tmp = t_2;
	elseif (z <= -8.2e-74)
		tmp = t_1;
	elseif (z <= 1e-39)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 1.55e+25)
		tmp = t_1;
	elseif (z <= 3.4e+101)
		tmp = t_2;
	else
		tmp = Float64(Float64(y / a) - Float64(Float64(x / z) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (z * a));
	t_2 = y / (a - (t / z));
	tmp = 0.0;
	if (z <= -4.4e+74)
		tmp = t_2;
	elseif (z <= -8.2e-74)
		tmp = t_1;
	elseif (z <= 1e-39)
		tmp = (x - (y * z)) / t;
	elseif (z <= 1.55e+25)
		tmp = t_1;
	elseif (z <= 3.4e+101)
		tmp = t_2;
	else
		tmp = (y / a) - ((x / z) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.4e+74], t$95$2, If[LessEqual[z, -8.2e-74], t$95$1, If[LessEqual[z, 1e-39], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.55e+25], t$95$1, If[LessEqual[z, 3.4e+101], t$95$2, N[(N[(y / a), $MachinePrecision] - N[(N[(x / z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.4000000000000002e74 or 1.5499999999999999e25 < z < 3.40000000000000017e101

   1. Initial program 67.8%

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

      1. *-commutative 67.8%

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

   3. Simplified 67.8%

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

      1. clear-num 67.1%

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

      2. inv-pow 67.1%

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

      3. sub-neg 67.1%

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

      4. +-commutative 67.1%

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

      5. *-commutative 67.1%

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

      6. distribute-rgt-neg-in 67.1%

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

      7. fma-def 67.1%

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

   5. Applied egg-rr 67.1%

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

   6. Taylor expanded in a around 0 66.7%

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

   7. Taylor expanded in y around inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. unsub-neg 89.3%

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

   9. Simplified 89.3%

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

   if -4.4000000000000002e74 < z < -8.20000000000000063e-74 or 9.99999999999999929e-40 < z < 1.5499999999999999e25

   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 x around inf 71.7%

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

      1. *-commutative 71.7%

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

   6. Simplified 71.7%

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

   if -8.20000000000000063e-74 < z < 9.99999999999999929e-40

   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 86.3%

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

   if 3.40000000000000017e101 < z

   1. Initial program 62.1%

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

      1. *-commutative 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in t around 0 53.7%

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

      1. associate-*r* 53.7%

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

      2. neg-mul-1 53.7%

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

      3. *-commutative 53.7%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in x around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. mul-1-neg 71.6%

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

      3. unsub-neg 71.6%

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

      4. *-commutative 71.6%

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

      5. associate-/r* 83.0%

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

   9. Simplified 83.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 10^{-39}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \end{array} \]

Alternative 4: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.66 \cdot 10^{+97}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.66e+97)
   (/ y (- a (/ t z)))
   (if (<= z 3.6e+149)
     (/ (- x (* y z)) (- t (* z a)))
     (- (/ y a) (/ (/ x z) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.66e+97) {
		tmp = y / (a - (t / z));
	} else if (z <= 3.6e+149) {
		tmp = (x - (y * z)) / (t - (z * a));
	} else {
		tmp = (y / a) - ((x / 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 <= (-1.66d+97)) then
        tmp = y / (a - (t / z))
    else if (z <= 3.6d+149) then
        tmp = (x - (y * z)) / (t - (z * a))
    else
        tmp = (y / a) - ((x / 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 <= -1.66e+97) {
		tmp = y / (a - (t / z));
	} else if (z <= 3.6e+149) {
		tmp = (x - (y * z)) / (t - (z * a));
	} else {
		tmp = (y / a) - ((x / z) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.66e+97:
		tmp = y / (a - (t / z))
	elif z <= 3.6e+149:
		tmp = (x - (y * z)) / (t - (z * a))
	else:
		tmp = (y / a) - ((x / z) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.66e+97)
		tmp = Float64(y / Float64(a - Float64(t / z)));
	elseif (z <= 3.6e+149)
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)));
	else
		tmp = Float64(Float64(y / a) - Float64(Float64(x / z) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.66e+97)
		tmp = y / (a - (t / z));
	elseif (z <= 3.6e+149)
		tmp = (x - (y * z)) / (t - (z * a));
	else
		tmp = (y / a) - ((x / z) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.66e+97], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e+149], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] - N[(N[(x / z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6599999999999999e97

   1. Initial program 59.4%

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

      1. *-commutative 59.4%

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

   3. Simplified 59.4%

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

      1. clear-num 59.4%

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

      2. inv-pow 59.4%

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

      3. sub-neg 59.4%

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

      4. +-commutative 59.4%

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

      5. *-commutative 59.4%

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

      6. distribute-rgt-neg-in 59.4%

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

      7. fma-def 59.4%

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

   5. Applied egg-rr 59.4%

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

   6. Taylor expanded in a around 0 58.8%

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

   7. Taylor expanded in y around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. unsub-neg 93.4%

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

   9. Simplified 93.4%

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

   if -1.6599999999999999e97 < z < 3.59999999999999995e149

   1. Initial program 97.8%

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

   if 3.59999999999999995e149 < z

   1. Initial program 55.6%

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

      1. *-commutative 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in t around 0 52.4%

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

      1. associate-*r* 52.4%

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

      2. neg-mul-1 52.4%

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

      3. *-commutative 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in x around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. *-commutative 75.1%

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

      5. associate-/r* 86.0%

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

   9. Simplified 86.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.66 \cdot 10^{+97}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \end{array} \]

Alternative 5: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-17}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ y (- a (/ t z)))))
   (if (<= z -5e+76)
     t_1
     (if (<= z -7.5e-76)
       (/ x (- t (* z a)))
       (if (<= z 1.32e-17) (/ (- x (* y z)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -5e+76) {
		tmp = t_1;
	} else if (z <= -7.5e-76) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.32e-17) {
		tmp = (x - (y * z)) / t;
	} 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) :: tmp
    t_1 = y / (a - (t / z))
    if (z <= (-5d+76)) then
        tmp = t_1
    else if (z <= (-7.5d-76)) then
        tmp = x / (t - (z * a))
    else if (z <= 1.32d-17) then
        tmp = (x - (y * z)) / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -5e+76) {
		tmp = t_1;
	} else if (z <= -7.5e-76) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.32e-17) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y / (a - (t / z))
	tmp = 0
	if z <= -5e+76:
		tmp = t_1
	elif z <= -7.5e-76:
		tmp = x / (t - (z * a))
	elif z <= 1.32e-17:
		tmp = (x - (y * z)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y / Float64(a - Float64(t / z)))
	tmp = 0.0
	if (z <= -5e+76)
		tmp = t_1;
	elseif (z <= -7.5e-76)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 1.32e-17)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y / (a - (t / z));
	tmp = 0.0;
	if (z <= -5e+76)
		tmp = t_1;
	elseif (z <= -7.5e-76)
		tmp = x / (t - (z * a));
	elseif (z <= 1.32e-17)
		tmp = (x - (y * z)) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5e+76], t$95$1, If[LessEqual[z, -7.5e-76], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.32e-17], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.99999999999999991e76 or 1.3200000000000001e-17 < z

   1. Initial program 67.5%

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

      1. *-commutative 67.5%

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

   3. Simplified 67.5%

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

      1. clear-num 67.1%

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

      2. inv-pow 67.1%

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

      3. sub-neg 67.1%

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

      4. +-commutative 67.1%

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

      5. *-commutative 67.1%

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

      6. distribute-rgt-neg-in 67.1%

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

      7. fma-def 67.1%

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

   5. Applied egg-rr 67.1%

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

   6. Taylor expanded in a around 0 65.7%

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

   7. Taylor expanded in y around inf 81.0%

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

      1. mul-1-neg 81.0%

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

      2. unsub-neg 81.0%

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

   9. Simplified 81.0%

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

   if -4.99999999999999991e76 < z < -7.4999999999999997e-76

   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. Taylor expanded in x around inf 66.0%

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

      1. *-commutative 66.0%

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

   6. Simplified 66.0%

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

   if -7.4999999999999997e-76 < z < 1.3200000000000001e-17

   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 85.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+76}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-17}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

Alternative 6: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{+79} \lor \neg \left(z \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.36e+79) (not (<= z 1.45e+26))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.36e+79) || !(z <= 1.45e+26)) {
		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 <= (-1.36d+79)) .or. (.not. (z <= 1.45d+26))) 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 <= -1.36e+79) || !(z <= 1.45e+26)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.36e+79) or not (z <= 1.45e+26):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.36e+79) || !(z <= 1.45e+26))
		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 <= -1.36e+79) || ~((z <= 1.45e+26)))
		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, -1.36e+79], N[Not[LessEqual[z, 1.45e+26]], $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 < -1.36000000000000003e79 or 1.45e26 < z

   1. Initial program 65.8%

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

      1. *-commutative 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in z around inf 60.8%

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

   if -1.36000000000000003e79 < z < 1.45e26

   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 x around inf 70.6%

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

      1. *-commutative 70.6%

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

   6. Simplified 70.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{+79} \lor \neg \left(z \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 7: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+86} \lor \neg \left(y \leq 6.2 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -7.2e+86) (not (<= y 6.2e-80)))
   (/ y (- a (/ t z)))
   (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -7.2e+86) || !(y <= 6.2e-80)) {
		tmp = y / (a - (t / z));
	} 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 <= (-7.2d+86)) .or. (.not. (y <= 6.2d-80))) then
        tmp = y / (a - (t / z))
    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 <= -7.2e+86) || !(y <= 6.2e-80)) {
		tmp = y / (a - (t / z));
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -7.2e+86) or not (y <= 6.2e-80):
		tmp = y / (a - (t / z))
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -7.2e+86) || !(y <= 6.2e-80))
		tmp = Float64(y / Float64(a - Float64(t / z)));
	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 <= -7.2e+86) || ~((y <= 6.2e-80)))
		tmp = y / (a - (t / z));
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -7.2e+86], N[Not[LessEqual[y, 6.2e-80]], $MachinePrecision]], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000011e86 or 6.20000000000000032e-80 < y

   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. Step-by-step derivation

      1. clear-num 80.6%

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

      2. inv-pow 80.6%

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

      3. sub-neg 80.6%

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

      4. +-commutative 80.6%

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

      5. *-commutative 80.6%

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

      6. distribute-rgt-neg-in 80.6%

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

      7. fma-def 80.6%

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

   5. Applied egg-rr 80.6%

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

   6. Taylor expanded in a around 0 78.7%

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

   7. Taylor expanded in y around inf 73.2%

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

      1. mul-1-neg 73.2%

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

      2. unsub-neg 73.2%

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

   9. Simplified 73.2%

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

   if -7.20000000000000011e86 < y < 6.20000000000000032e-80

   1. Initial program 93.2%

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

      1. *-commutative 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in x around inf 78.2%

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

      1. *-commutative 78.2%

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

   6. Simplified 78.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+86} \lor \neg \left(y \leq 6.2 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 8: 54.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+54}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -9.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 10^{-15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.26e+54)
   (/ y a)
   (if (<= z -9.4e-42) (/ (- z) (/ t y)) (if (<= z 1e-15) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.26e+54) {
		tmp = y / a;
	} else if (z <= -9.4e-42) {
		tmp = -z / (t / y);
	} else if (z <= 1e-15) {
		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 <= (-1.26d+54)) then
        tmp = y / a
    else if (z <= (-9.4d-42)) then
        tmp = -z / (t / y)
    else if (z <= 1d-15) 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 <= -1.26e+54) {
		tmp = y / a;
	} else if (z <= -9.4e-42) {
		tmp = -z / (t / y);
	} else if (z <= 1e-15) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.26e+54:
		tmp = y / a
	elif z <= -9.4e-42:
		tmp = -z / (t / y)
	elif z <= 1e-15:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.26e+54)
		tmp = Float64(y / a);
	elseif (z <= -9.4e-42)
		tmp = Float64(Float64(-z) / Float64(t / y));
	elseif (z <= 1e-15)
		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 <= -1.26e+54)
		tmp = y / a;
	elseif (z <= -9.4e-42)
		tmp = -z / (t / y);
	elseif (z <= 1e-15)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.26e+54], N[(y / a), $MachinePrecision], If[LessEqual[z, -9.4e-42], N[((-z) / N[(t / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-15], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25999999999999995e54 or 1.0000000000000001e-15 < z

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in z around inf 58.7%

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

   if -1.25999999999999995e54 < z < -9.4000000000000001e-42

   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 x around 0 39.8%

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

      1. mul-1-neg 39.8%

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

      2. associate-/l* 39.8%

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

      3. *-commutative 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in t around inf 33.1%

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

      1. *-commutative 33.1%

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

      2. associate-/l* 36.7%

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

   9. Simplified 36.7%

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

   if -9.4000000000000001e-42 < z < 1.0000000000000001e-15

   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 61.7%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+54}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -9.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 10^{-15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9: 54.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+54}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.45 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.25e+54)
   (/ y a)
   (if (<= z -3.45e-41)
     (/ (/ (- x) a) z)
     (if (<= z 2.75e-15) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e+54) {
		tmp = y / a;
	} else if (z <= -3.45e-41) {
		tmp = (-x / a) / z;
	} else if (z <= 2.75e-15) {
		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+54)) then
        tmp = y / a
    else if (z <= (-3.45d-41)) then
        tmp = (-x / a) / z
    else if (z <= 2.75d-15) 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+54) {
		tmp = y / a;
	} else if (z <= -3.45e-41) {
		tmp = (-x / a) / z;
	} else if (z <= 2.75e-15) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.25e+54:
		tmp = y / a
	elif z <= -3.45e-41:
		tmp = (-x / a) / z
	elif z <= 2.75e-15:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.25e+54)
		tmp = Float64(y / a);
	elseif (z <= -3.45e-41)
		tmp = Float64(Float64(Float64(-x) / a) / z);
	elseif (z <= 2.75e-15)
		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+54)
		tmp = y / a;
	elseif (z <= -3.45e-41)
		tmp = (-x / a) / z;
	elseif (z <= 2.75e-15)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.25e+54], N[(y / a), $MachinePrecision], If[LessEqual[z, -3.45e-41], N[(N[((-x) / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 2.75e-15], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25000000000000001e54 or 2.7500000000000001e-15 < z

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in z around inf 58.7%

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

   if -1.25000000000000001e54 < z < -3.4499999999999999e-41

   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 0 70.5%

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

      1. associate-*r* 70.5%

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

      2. neg-mul-1 70.5%

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

      3. *-commutative 70.5%

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

   6. Simplified 70.5%

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

      1. *-un-lft-identity 70.5%

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

      2. times-frac 68.1%

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

      3. frac-2neg 68.1%

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

      4. metadata-eval 68.1%

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

      5. add-sqr-sqrt 67.8%

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

      6. sqrt-unprod 68.1%

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

      7. sqr-neg 68.1%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 5.6%

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

      10. sub-neg 5.6%

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

      11. *-commutative 5.6%

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

      12. distribute-lft-neg-in 5.6%

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

      13. add-sqr-sqrt 5.6%

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

      14. sqrt-unprod 5.6%

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

      15. sqr-neg 5.6%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 11.5%

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

      18. add-sqr-sqrt 6.3%

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

      19. sqrt-unprod 26.2%

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

      20. sqr-neg 26.2%

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

      21. sqrt-unprod 27.2%

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

      22. add-sqr-sqrt 62.2%

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

   8. Applied egg-rr 62.2%

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

   9. Taylor expanded in z around 0 62.9%

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

       1. mul-1-neg 62.9%

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

       2. associate-/r* 62.7%

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

   11. Simplified 62.7%

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

   if -3.4499999999999999e-41 < z < 2.7500000000000001e-15

   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 61.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+54}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.45 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10: 54.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+53}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.45 \cdot 10^{-41}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.1e+53)
   (/ y a)
   (if (<= z -3.45e-41)
     (/ (- x) (* z a))
     (if (<= z 2.85e-15) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.1e+53) {
		tmp = y / a;
	} else if (z <= -3.45e-41) {
		tmp = -x / (z * a);
	} else if (z <= 2.85e-15) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.1d+53)) then
        tmp = y / a
    else if (z <= (-3.45d-41)) then
        tmp = -x / (z * a)
    else if (z <= 2.85d-15) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.1e+53) {
		tmp = y / a;
	} else if (z <= -3.45e-41) {
		tmp = -x / (z * a);
	} else if (z <= 2.85e-15) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.1e+53:
		tmp = y / a
	elif z <= -3.45e-41:
		tmp = -x / (z * a)
	elif z <= 2.85e-15:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.1e+53)
		tmp = Float64(y / a);
	elseif (z <= -3.45e-41)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 2.85e-15)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.1e+53)
		tmp = y / a;
	elseif (z <= -3.45e-41)
		tmp = -x / (z * a);
	elseif (z <= 2.85e-15)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.1e+53], N[(y / a), $MachinePrecision], If[LessEqual[z, -3.45e-41], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.85e-15], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.0999999999999998e53 or 2.8500000000000002e-15 < z

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in z around inf 58.7%

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

   if -5.0999999999999998e53 < z < -3.4499999999999999e-41

   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 0 70.5%

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

      1. associate-*r* 70.5%

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

      2. neg-mul-1 70.5%

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

      3. *-commutative 70.5%

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

   6. Simplified 70.5%

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

      1. clear-num 70.3%

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

      2. associate-/r/ 70.3%

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

      3. associate-/r* 70.2%

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

      4. frac-2neg 70.2%

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

      5. metadata-eval 70.2%

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

      6. add-sqr-sqrt 70.0%

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

      7. sqrt-unprod 70.2%

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

      8. sqr-neg 70.2%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 5.6%

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

      11. add-sqr-sqrt 1.2%

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

      12. sqrt-unprod 25.3%

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

      13. sqr-neg 25.3%

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

      14. sqrt-unprod 27.9%

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

      15. add-sqr-sqrt 70.2%

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

      16. sub-neg 70.2%

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

      17. *-commutative 70.2%

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

      18. distribute-lft-neg-in 70.2%

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

      19. add-sqr-sqrt 70.2%

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

      20. sqrt-unprod 70.2%

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

      21. sqr-neg 70.2%

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

      22. sqrt-unprod 0.0%

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

      23. add-sqr-sqrt 62.2%

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

   8. Applied egg-rr 62.2%

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

   9. Taylor expanded in z around 0 62.9%

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

       1. associate-*r/ 62.9%

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

       2. neg-mul-1 62.9%

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

   11. Simplified 62.9%

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

   if -3.4499999999999999e-41 < z < 2.8500000000000002e-15

   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 61.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+53}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.45 \cdot 10^{-41}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 11: 53.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+75} \lor \neg \left(z \leq 8.2 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6.3e+75) (not (<= z 8.2e-18))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6.3e+75) || !(z <= 8.2e-18)) {
		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 <= (-6.3d+75)) .or. (.not. (z <= 8.2d-18))) 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 <= -6.3e+75) || !(z <= 8.2e-18)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -6.3e+75) or not (z <= 8.2e-18):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6.3e+75) || !(z <= 8.2e-18))
		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 <= -6.3e+75) || ~((z <= 8.2e-18)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6.3e+75], N[Not[LessEqual[z, 8.2e-18]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.30000000000000036e75 or 8.1999999999999995e-18 < z

   1. Initial program 67.5%

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

      1. *-commutative 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in z around inf 59.9%

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

   if -6.30000000000000036e75 < z < 8.1999999999999995e-18

   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 53.5%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+75} \lor \neg \left(z \leq 8.2 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 12: 34.2% 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 87.1%

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

   1. *-commutative 87.1%

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

3. Simplified 87.1%

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

4. Taylor expanded in z around 0 36.3%

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

5. Final simplification 36.3%

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

Developer target: 97.0% 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 2023332 
(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))))