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

Percentage Accurate: 85.1% → 94.2%

Time: 16.7s

Alternatives: 13

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, -a, t\right)\\ t_2 := \mathsf{fma}\left(-1, y \cdot \frac{z}{t\_1}, \frac{x}{t\_1}\right)\\ t_3 := t - z \cdot a\\ t_4 := \frac{x - y \cdot z}{t\_3}\\ \mathbf{if}\;t\_4 \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{t\_3} + \frac{y \cdot z}{z \cdot a - t}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (- a) t))
        (t_2 (fma -1.0 (* y (/ z t_1)) (/ x t_1)))
        (t_3 (- t (* z a)))
        (t_4 (/ (- x (* y z)) t_3)))
   (if (<= t_4 -2.8e-10)
     t_2
     (if (<= t_4 4e+23)
       (+ (/ x t_3) (/ (* y z) (- (* z a) t)))
       (if (<= t_4 INFINITY) t_2 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, -a, t);
	double t_2 = fma(-1.0, (y * (z / t_1)), (x / t_1));
	double t_3 = t - (z * a);
	double t_4 = (x - (y * z)) / t_3;
	double tmp;
	if (t_4 <= -2.8e-10) {
		tmp = t_2;
	} else if (t_4 <= 4e+23) {
		tmp = (x / t_3) + ((y * z) / ((z * a) - t));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(-a), t)
	t_2 = fma(-1.0, Float64(y * Float64(z / t_1)), Float64(x / t_1))
	t_3 = Float64(t - Float64(z * a))
	t_4 = Float64(Float64(x - Float64(y * z)) / t_3)
	tmp = 0.0
	if (t_4 <= -2.8e-10)
		tmp = t_2;
	elseif (t_4 <= 4e+23)
		tmp = Float64(Float64(x / t_3) + Float64(Float64(y * z) / Float64(Float64(z * a) - t)));
	elseif (t_4 <= 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[(z * (-a) + t), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -2.8e-10], t$95$2, If[LessEqual[t$95$4, 4e+23], N[(N[(x / t$95$3), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 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))) < -2.80000000000000015e-10 or 3.9999999999999997e23 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in x around 0 85.6%

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

      1. fma-define 85.6%

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

      2. associate-/l* 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. cancel-sign-sub-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-define 99.7%

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

   7. Simplified 99.7%

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

   if -2.80000000000000015e-10 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 3.9999999999999997e23

   1. Initial program 91.7%

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

      1. *-commutative 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in x around 0 91.7%

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

      1. fma-define 91.7%

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

      2. associate-/l* 85.9%

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

      3. cancel-sign-sub-inv 85.9%

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

      4. *-commutative 85.9%

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

      5. +-commutative 85.9%

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

      6. fma-define 85.9%

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

      7. cancel-sign-sub-inv 85.9%

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

      8. *-commutative 85.9%

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

      9. +-commutative 85.9%

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

      10. fma-define 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in y around 0 91.7%

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

      1. +-commutative 91.7%

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

      2. mul-1-neg 91.7%

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

      3. unsub-neg 91.7%

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

      4. mul-1-neg 91.7%

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

      5. sub-neg 91.7%

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

      6. *-commutative 91.7%

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

      7. associate-/l* 85.9%

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

      8. mul-1-neg 85.9%

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

      9. sub-neg 85.9%

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

      10. *-commutative 85.9%

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

   10. Simplified 85.9%

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

       1. clear-num 85.3%

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

       2. inv-pow 85.3%

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

   12. Applied egg-rr 85.3%

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

       1. unpow-1 85.3%

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

   14. Simplified 85.3%

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

   15. Taylor expanded in y around 0 91.7%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around inf 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-1, y \cdot \frac{z}{\mathsf{fma}\left(z, -a, t\right)}, \frac{x}{\mathsf{fma}\left(z, -a, t\right)}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{t - z \cdot a} + \frac{y \cdot z}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-1, y \cdot \frac{z}{\mathsf{fma}\left(z, -a, t\right)}, \frac{x}{\mathsf{fma}\left(z, -a, t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x}{t\_1} + y \cdot \frac{z}{z \cdot a - t}\\ t_3 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-58}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a)))
        (t_2 (+ (/ x t_1) (* y (/ z (- (* z a) t)))))
        (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 -2.8e-10)
     t_2
     (if (<= t_3 5e-58) t_3 (if (<= t_3 INFINITY) t_2 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x / t_1) + (y * (z / ((z * a) - t)));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -2.8e-10) {
		tmp = t_2;
	} else if (t_3 <= 5e-58) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x / t_1) + (y * (z / ((z * a) - t)));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -2.8e-10) {
		tmp = t_2;
	} else if (t_3 <= 5e-58) {
		tmp = t_3;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = (x / t_1) + (y * (z / ((z * a) - t)))
	t_3 = (x - (y * z)) / t_1
	tmp = 0
	if t_3 <= -2.8e-10:
		tmp = t_2
	elif t_3 <= 5e-58:
		tmp = t_3
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(x / t_1) + Float64(y * Float64(z / Float64(Float64(z * a) - t))))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= -2.8e-10)
		tmp = t_2;
	elseif (t_3 <= 5e-58)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = (x / t_1) + (y * (z / ((z * a) - t)));
	t_3 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_3 <= -2.8e-10)
		tmp = t_2;
	elseif (t_3 <= 5e-58)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] + N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -2.8e-10], t$95$2, If[LessEqual[t$95$3, 5e-58], t$95$3, 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))) < -2.80000000000000015e-10 or 4.99999999999999977e-58 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

   3. Simplified 87.3%

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

   5. Taylor expanded in x around 0 87.3%

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

      1. fma-define 87.3%

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

      2. associate-/l* 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. cancel-sign-sub-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-define 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0 87.3%

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

      1. +-commutative 87.3%

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

      2. mul-1-neg 87.3%

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

      3. unsub-neg 87.3%

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

      4. mul-1-neg 87.3%

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

      5. sub-neg 87.3%

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

      6. *-commutative 87.3%

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

      7. associate-/l* 99.7%

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

      8. mul-1-neg 99.7%

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

      9. sub-neg 99.7%

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

      10. *-commutative 99.7%

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

   10. Simplified 99.7%

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

   if -2.80000000000000015e-10 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.99999999999999977e-58

   1. Initial program 90.5%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around inf 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{t - z \cdot a} + y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x}{t - z \cdot a} + y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot a - t\\ t_2 := t - z \cdot a\\ t_3 := \frac{x}{t\_2}\\ t_4 := t\_3 + y \cdot \frac{z}{t\_1}\\ t_5 := \frac{x - y \cdot z}{t\_2}\\ \mathbf{if}\;t\_5 \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-58}:\\ \;\;\;\;t\_3 + \frac{y \cdot z}{t\_1}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* z a) t))
        (t_2 (- t (* z a)))
        (t_3 (/ x t_2))
        (t_4 (+ t_3 (* y (/ z t_1))))
        (t_5 (/ (- x (* y z)) t_2)))
   (if (<= t_5 -2.8e-10)
     t_4
     (if (<= t_5 5e-58)
       (+ t_3 (/ (* y z) t_1))
       (if (<= t_5 INFINITY) t_4 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * a) - t;
	double t_2 = t - (z * a);
	double t_3 = x / t_2;
	double t_4 = t_3 + (y * (z / t_1));
	double t_5 = (x - (y * z)) / t_2;
	double tmp;
	if (t_5 <= -2.8e-10) {
		tmp = t_4;
	} else if (t_5 <= 5e-58) {
		tmp = t_3 + ((y * z) / t_1);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * a) - t;
	double t_2 = t - (z * a);
	double t_3 = x / t_2;
	double t_4 = t_3 + (y * (z / t_1));
	double t_5 = (x - (y * z)) / t_2;
	double tmp;
	if (t_5 <= -2.8e-10) {
		tmp = t_4;
	} else if (t_5 <= 5e-58) {
		tmp = t_3 + ((y * z) / t_1);
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = t_4;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * a) - t
	t_2 = t - (z * a)
	t_3 = x / t_2
	t_4 = t_3 + (y * (z / t_1))
	t_5 = (x - (y * z)) / t_2
	tmp = 0
	if t_5 <= -2.8e-10:
		tmp = t_4
	elif t_5 <= 5e-58:
		tmp = t_3 + ((y * z) / t_1)
	elif t_5 <= math.inf:
		tmp = t_4
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * a) - t)
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64(x / t_2)
	t_4 = Float64(t_3 + Float64(y * Float64(z / t_1)))
	t_5 = Float64(Float64(x - Float64(y * z)) / t_2)
	tmp = 0.0
	if (t_5 <= -2.8e-10)
		tmp = t_4;
	elseif (t_5 <= 5e-58)
		tmp = Float64(t_3 + Float64(Float64(y * z) / t_1));
	elseif (t_5 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * a) - t;
	t_2 = t - (z * a);
	t_3 = x / t_2;
	t_4 = t_3 + (y * (z / t_1));
	t_5 = (x - (y * z)) / t_2;
	tmp = 0.0;
	if (t_5 <= -2.8e-10)
		tmp = t_4;
	elseif (t_5 <= 5e-58)
		tmp = t_3 + ((y * z) / t_1);
	elseif (t_5 <= Inf)
		tmp = t_4;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$5, -2.8e-10], t$95$4, If[LessEqual[t$95$5, 5e-58], N[(t$95$3 + N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$4, N[(y / a), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -2.80000000000000015e-10 or 4.99999999999999977e-58 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

   3. Simplified 87.3%

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

   5. Taylor expanded in x around 0 87.3%

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

      1. fma-define 87.3%

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

      2. associate-/l* 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. cancel-sign-sub-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-define 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0 87.3%

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

      1. +-commutative 87.3%

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

      2. mul-1-neg 87.3%

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

      3. unsub-neg 87.3%

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

      4. mul-1-neg 87.3%

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

      5. sub-neg 87.3%

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

      6. *-commutative 87.3%

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

      7. associate-/l* 99.7%

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

      8. mul-1-neg 99.7%

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

      9. sub-neg 99.7%

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

      10. *-commutative 99.7%

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

   10. Simplified 99.7%

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

   if -2.80000000000000015e-10 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.99999999999999977e-58

   1. Initial program 90.5%

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

      1. *-commutative 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in x around 0 90.6%

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

      1. fma-define 90.6%

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

      2. associate-/l* 83.9%

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

      3. cancel-sign-sub-inv 83.9%

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

      4. *-commutative 83.9%

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

      5. +-commutative 83.9%

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

      6. fma-define 83.9%

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

      7. cancel-sign-sub-inv 83.9%

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

      8. *-commutative 83.9%

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

      9. +-commutative 83.9%

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

      10. fma-define 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in y around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. mul-1-neg 90.6%

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

      3. unsub-neg 90.6%

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

      4. mul-1-neg 90.6%

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

      5. sub-neg 90.6%

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

      6. *-commutative 90.6%

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

      7. associate-/l* 83.9%

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

      8. mul-1-neg 83.9%

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

      9. sub-neg 83.9%

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

      10. *-commutative 83.9%

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

   10. Simplified 83.9%

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

       1. clear-num 83.2%

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

       2. inv-pow 83.2%

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

   12. Applied egg-rr 83.2%

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

       1. unpow-1 83.2%

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

   14. Simplified 83.2%

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

   15. Taylor expanded in y around 0 90.6%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around inf 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{t - z \cdot a} + y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{t - z \cdot a} + \frac{y \cdot z}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x}{t - z \cdot a} + y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 (- INFINITY))
     (* y (/ z (- (* z a) t)))
     (if (<= t_1 2e+305) t_1 (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (z / ((z * a) - t));
	} else if (t_1 <= 2e+305) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (z / ((z * a) - t));
	} else if (t_1 <= 2e+305) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * (z / ((z * a) - t))
	elif t_1 <= 2e+305:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / Float64(Float64(z * a) - t)));
	elseif (t_1 <= 2e+305)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * (z / ((z * a) - t));
	elseif (t_1 <= 2e+305)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+305], t$95$1, 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 47.2%

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

      1. *-commutative 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in x around 0 47.2%

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

      1. fma-define 47.2%

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

      2. associate-/l* 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. cancel-sign-sub-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-define 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 19.8%

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

      1. mul-1-neg 19.8%

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

      2. associate-/l* 72.3%

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

      3. distribute-rgt-neg-in 72.3%

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

      4. mul-1-neg 72.3%

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

      5. sub-neg 72.3%

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

      6. *-commutative 72.3%

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

   10. Simplified 72.3%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.9999999999999999e305

   1. Initial program 95.0%

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

   if 1.9999999999999999e305 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 29.9%

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

      1. *-commutative 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in z around inf 84.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{t} - y \cdot \frac{-1}{\frac{z \cdot a - t}{z}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 (- INFINITY))
     (- (/ x t) (* y (/ -1.0 (/ (- (* z a) t) z))))
     (if (<= t_1 2e+305) t_1 (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x / t) - (y * (-1.0 / (((z * a) - t) / z)));
	} else if (t_1 <= 2e+305) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x / t) - (y * (-1.0 / (((z * a) - t) / z)));
	} else if (t_1 <= 2e+305) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x / t) - (y * (-1.0 / (((z * a) - t) / z)))
	elif t_1 <= 2e+305:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x / t) - Float64(y * Float64(-1.0 / Float64(Float64(Float64(z * a) - t) / z))));
	elseif (t_1 <= 2e+305)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x / t) - (y * (-1.0 / (((z * a) - t) / z)));
	elseif (t_1 <= 2e+305)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x / t), $MachinePrecision] - N[(y * N[(-1.0 / N[(N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+305], t$95$1, 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 47.2%

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

      1. *-commutative 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in x around 0 47.2%

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

      1. fma-define 47.2%

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

      2. associate-/l* 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. cancel-sign-sub-inv 99.7%

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

      8. *-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-define 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0 47.2%

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

      1. +-commutative 47.2%

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

      2. mul-1-neg 47.2%

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

      3. unsub-neg 47.2%

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

      4. mul-1-neg 47.2%

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

      5. sub-neg 47.2%

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

      6. *-commutative 47.2%

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

      7. associate-/l* 99.7%

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

      8. mul-1-neg 99.7%

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

      9. sub-neg 99.7%

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

      10. *-commutative 99.7%

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

   10. Simplified 99.7%

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

       1. clear-num 99.7%

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

       2. inv-pow 99.7%

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

   12. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

   14. Simplified 99.7%

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

   15. Taylor expanded in t around inf 72.7%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.9999999999999999e305

   1. Initial program 95.0%

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

   if 1.9999999999999999e305 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 29.9%

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

      1. *-commutative 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in z around inf 84.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{x}{t} - y \cdot \frac{-1}{\frac{z \cdot a - t}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{-y}{t}\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.18 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 10^{-73}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ (- y) t))))
   (if (<= z -6.2e+87)
     (/ y a)
     (if (<= z -1.18e+27)
       t_1
       (if (<= z -9.5e-18)
         (/ y a)
         (if (<= z 1e-73) (/ x t) (if (<= z 2e+82) t_1 (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (-y / t);
	double tmp;
	if (z <= -6.2e+87) {
		tmp = y / a;
	} else if (z <= -1.18e+27) {
		tmp = t_1;
	} else if (z <= -9.5e-18) {
		tmp = y / a;
	} else if (z <= 1e-73) {
		tmp = x / t;
	} else if (z <= 2e+82) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (-y / t)
    if (z <= (-6.2d+87)) then
        tmp = y / a
    else if (z <= (-1.18d+27)) then
        tmp = t_1
    else if (z <= (-9.5d-18)) then
        tmp = y / a
    else if (z <= 1d-73) then
        tmp = x / t
    else if (z <= 2d+82) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (-y / t);
	double tmp;
	if (z <= -6.2e+87) {
		tmp = y / a;
	} else if (z <= -1.18e+27) {
		tmp = t_1;
	} else if (z <= -9.5e-18) {
		tmp = y / a;
	} else if (z <= 1e-73) {
		tmp = x / t;
	} else if (z <= 2e+82) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * (-y / t)
	tmp = 0
	if z <= -6.2e+87:
		tmp = y / a
	elif z <= -1.18e+27:
		tmp = t_1
	elif z <= -9.5e-18:
		tmp = y / a
	elif z <= 1e-73:
		tmp = x / t
	elif z <= 2e+82:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(Float64(-y) / t))
	tmp = 0.0
	if (z <= -6.2e+87)
		tmp = Float64(y / a);
	elseif (z <= -1.18e+27)
		tmp = t_1;
	elseif (z <= -9.5e-18)
		tmp = Float64(y / a);
	elseif (z <= 1e-73)
		tmp = Float64(x / t);
	elseif (z <= 2e+82)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * (-y / t);
	tmp = 0.0;
	if (z <= -6.2e+87)
		tmp = y / a;
	elseif (z <= -1.18e+27)
		tmp = t_1;
	elseif (z <= -9.5e-18)
		tmp = y / a;
	elseif (z <= 1e-73)
		tmp = x / t;
	elseif (z <= 2e+82)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[((-y) / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.2e+87], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.18e+27], t$95$1, If[LessEqual[z, -9.5e-18], N[(y / a), $MachinePrecision], If[LessEqual[z, 1e-73], N[(x / t), $MachinePrecision], If[LessEqual[z, 2e+82], t$95$1, N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.1999999999999999e87 or -1.18000000000000006e27 < z < -9.5000000000000003e-18 or 1.9999999999999999e82 < z

   1. Initial program 67.0%

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

      1. *-commutative 67.0%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in z around inf 58.1%

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

   if -6.1999999999999999e87 < z < -1.18000000000000006e27 or 9.99999999999999997e-74 < z < 1.9999999999999999e82

   1. Initial program 95.4%

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

      1. *-commutative 95.4%

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

   3. Simplified 95.4%

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

      1. clear-num 95.3%

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

      2. associate-/r/ 95.1%

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

      3. sub-neg 95.1%

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

      4. +-commutative 95.1%

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

      5. *-commutative 95.1%

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

      6. distribute-rgt-neg-in 95.1%

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

      7. fma-define 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in a around 0 47.1%

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

   8. Taylor expanded in x around 0 39.3%

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

      1. mul-1-neg 39.3%

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

      2. *-commutative 39.3%

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

      3. associate-/l* 45.6%

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

      4. distribute-lft-neg-in 45.6%

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

   10. Simplified 45.6%

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

   if -9.5000000000000003e-18 < z < 9.99999999999999997e-74

   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. Add Preprocessing

   5. Taylor expanded in z around 0 54.9%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.18 \cdot 10^{+27}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 10^{-73}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+82}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;a \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{-45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq 1.52 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) t)) (t_2 (/ (- y (/ x z)) a)))
   (if (<= a -5e+18)
     t_2
     (if (<= a -2.5e-34)
       t_1
       (if (<= a -5.3e-45)
         t_2
         (if (<= a 1.52e-46) t_1 (- (/ y a) (/ (/ x a) z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / t;
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (a <= -5e+18) {
		tmp = t_2;
	} else if (a <= -2.5e-34) {
		tmp = t_1;
	} else if (a <= -5.3e-45) {
		tmp = t_2;
	} else if (a <= 1.52e-46) {
		tmp = t_1;
	} else {
		tmp = (y / a) - ((x / a) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x - (y * z)) / t
    t_2 = (y - (x / z)) / a
    if (a <= (-5d+18)) then
        tmp = t_2
    else if (a <= (-2.5d-34)) then
        tmp = t_1
    else if (a <= (-5.3d-45)) then
        tmp = t_2
    else if (a <= 1.52d-46) then
        tmp = t_1
    else
        tmp = (y / a) - ((x / a) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / t;
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (a <= -5e+18) {
		tmp = t_2;
	} else if (a <= -2.5e-34) {
		tmp = t_1;
	} else if (a <= -5.3e-45) {
		tmp = t_2;
	} else if (a <= 1.52e-46) {
		tmp = t_1;
	} else {
		tmp = (y / a) - ((x / a) / z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / t
	t_2 = (y - (x / z)) / a
	tmp = 0
	if a <= -5e+18:
		tmp = t_2
	elif a <= -2.5e-34:
		tmp = t_1
	elif a <= -5.3e-45:
		tmp = t_2
	elif a <= 1.52e-46:
		tmp = t_1
	else:
		tmp = (y / a) - ((x / a) / z)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / t)
	t_2 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (a <= -5e+18)
		tmp = t_2;
	elseif (a <= -2.5e-34)
		tmp = t_1;
	elseif (a <= -5.3e-45)
		tmp = t_2;
	elseif (a <= 1.52e-46)
		tmp = t_1;
	else
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / t;
	t_2 = (y - (x / z)) / a;
	tmp = 0.0;
	if (a <= -5e+18)
		tmp = t_2;
	elseif (a <= -2.5e-34)
		tmp = t_1;
	elseif (a <= -5.3e-45)
		tmp = t_2;
	elseif (a <= 1.52e-46)
		tmp = t_1;
	else
		tmp = (y / a) - ((x / a) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[a, -5e+18], t$95$2, If[LessEqual[a, -2.5e-34], t$95$1, If[LessEqual[a, -5.3e-45], t$95$2, If[LessEqual[a, 1.52e-46], t$95$1, N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5e18 or -2.5000000000000001e-34 < a < -5.2999999999999997e-45

   1. Initial program 75.2%

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

      1. *-commutative 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in x around 0 75.2%

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

      1. fma-define 75.2%

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

      2. associate-/l* 83.4%

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

      3. cancel-sign-sub-inv 83.4%

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

      4. *-commutative 83.4%

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

      5. +-commutative 83.4%

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

      6. fma-define 83.5%

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

      7. cancel-sign-sub-inv 83.5%

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

      8. *-commutative 83.5%

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

      9. +-commutative 83.5%

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

      10. fma-define 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in a around inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. unsub-neg 80.1%

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

   10. Simplified 80.1%

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

   if -5e18 < a < -2.5000000000000001e-34 or -5.2999999999999997e-45 < a < 1.52000000000000006e-46

   1. Initial program 95.2%

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

      1. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in t around inf 79.6%

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

   if 1.52000000000000006e-46 < a

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in t around 0 58.6%

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

      1. associate-*r/ 58.6%

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

      2. neg-mul-1 58.6%

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

      3. neg-sub0 58.6%

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

      4. sub-neg 58.6%

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

      5. distribute-rgt-neg-out 58.6%

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

      6. +-commutative 58.6%

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

      7. associate--r+ 58.6%

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

      8. neg-sub0 58.6%

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

      9. distribute-rgt-neg-out 58.6%

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

      10. remove-double-neg 58.6%

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

      11. *-commutative 58.6%

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

   7. Simplified 58.6%

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

      1. clear-num 58.5%

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

      2. inv-pow 58.5%

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

      3. *-commutative 58.5%

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

      4. fma-neg 58.5%

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

   9. Applied egg-rr 58.5%

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

       1. unpow-1 58.5%

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

       2. fma-neg 58.5%

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

   11. Simplified 58.5%

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

   12. Taylor expanded in z around 0 75.5%

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

       1. +-commutative 75.5%

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

       2. mul-1-neg 75.5%

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

       3. unsub-neg 75.5%

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

       4. associate-/r* 76.0%

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

   14. Simplified 76.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;a \leq 1.52 \cdot 10^{-46}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+263}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq -2.95 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -3900000000000:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= a -3.8e+263)
     (/ y a)
     (if (<= a -2.95e+103)
       t_1
       (if (<= a -3900000000000.0)
         (/ y a)
         (if (<= a 6.5e-53) (/ (- x (* y z)) t) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (a <= -3.8e+263) {
		tmp = y / a;
	} else if (a <= -2.95e+103) {
		tmp = t_1;
	} else if (a <= -3900000000000.0) {
		tmp = y / a;
	} else if (a <= 6.5e-53) {
		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 = x / (t - (z * a))
    if (a <= (-3.8d+263)) then
        tmp = y / a
    else if (a <= (-2.95d+103)) then
        tmp = t_1
    else if (a <= (-3900000000000.0d0)) then
        tmp = y / a
    else if (a <= 6.5d-53) 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 = x / (t - (z * a));
	double tmp;
	if (a <= -3.8e+263) {
		tmp = y / a;
	} else if (a <= -2.95e+103) {
		tmp = t_1;
	} else if (a <= -3900000000000.0) {
		tmp = y / a;
	} else if (a <= 6.5e-53) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	tmp = 0
	if a <= -3.8e+263:
		tmp = y / a
	elif a <= -2.95e+103:
		tmp = t_1
	elif a <= -3900000000000.0:
		tmp = y / a
	elif a <= 6.5e-53:
		tmp = (x - (y * z)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (a <= -3.8e+263)
		tmp = Float64(y / a);
	elseif (a <= -2.95e+103)
		tmp = t_1;
	elseif (a <= -3900000000000.0)
		tmp = Float64(y / a);
	elseif (a <= 6.5e-53)
		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 = x / (t - (z * a));
	tmp = 0.0;
	if (a <= -3.8e+263)
		tmp = y / a;
	elseif (a <= -2.95e+103)
		tmp = t_1;
	elseif (a <= -3900000000000.0)
		tmp = y / a;
	elseif (a <= 6.5e-53)
		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[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.8e+263], N[(y / a), $MachinePrecision], If[LessEqual[a, -2.95e+103], t$95$1, If[LessEqual[a, -3900000000000.0], N[(y / a), $MachinePrecision], If[LessEqual[a, 6.5e-53], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7999999999999999e263 or -2.9499999999999999e103 < a < -3.9e12

   1. Initial program 62.7%

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

      1. *-commutative 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in z around inf 78.2%

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

   if -3.7999999999999999e263 < a < -2.9499999999999999e103 or 6.4999999999999997e-53 < a

   1. Initial program 79.5%

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

      1. *-commutative 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around inf 58.4%

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

   if -3.9e12 < a < 6.4999999999999997e-53

   1. Initial program 94.6%

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

      1. *-commutative 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in t around inf 76.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+263}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq -2.95 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;a \leq -3900000000000:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+129}:\\ \;\;\;\;\frac{y \cdot z}{-t}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+207}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.02e+27)
   (/ y a)
   (if (<= y 1.3e+89)
     (/ x (- t (* z a)))
     (if (<= y 1.55e+129)
       (/ (* y z) (- t))
       (if (<= y 2.5e+207) (/ y a) (* z (/ (- y) t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.02e+27) {
		tmp = y / a;
	} else if (y <= 1.3e+89) {
		tmp = x / (t - (z * a));
	} else if (y <= 1.55e+129) {
		tmp = (y * z) / -t;
	} else if (y <= 2.5e+207) {
		tmp = y / a;
	} else {
		tmp = z * (-y / 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 (y <= (-1.02d+27)) then
        tmp = y / a
    else if (y <= 1.3d+89) then
        tmp = x / (t - (z * a))
    else if (y <= 1.55d+129) then
        tmp = (y * z) / -t
    else if (y <= 2.5d+207) then
        tmp = y / a
    else
        tmp = z * (-y / 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 (y <= -1.02e+27) {
		tmp = y / a;
	} else if (y <= 1.3e+89) {
		tmp = x / (t - (z * a));
	} else if (y <= 1.55e+129) {
		tmp = (y * z) / -t;
	} else if (y <= 2.5e+207) {
		tmp = y / a;
	} else {
		tmp = z * (-y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -1.02e+27:
		tmp = y / a
	elif y <= 1.3e+89:
		tmp = x / (t - (z * a))
	elif y <= 1.55e+129:
		tmp = (y * z) / -t
	elif y <= 2.5e+207:
		tmp = y / a
	else:
		tmp = z * (-y / t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.02e+27)
		tmp = Float64(y / a);
	elseif (y <= 1.3e+89)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (y <= 1.55e+129)
		tmp = Float64(Float64(y * z) / Float64(-t));
	elseif (y <= 2.5e+207)
		tmp = Float64(y / a);
	else
		tmp = Float64(z * Float64(Float64(-y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -1.02e+27)
		tmp = y / a;
	elseif (y <= 1.3e+89)
		tmp = x / (t - (z * a));
	elseif (y <= 1.55e+129)
		tmp = (y * z) / -t;
	elseif (y <= 2.5e+207)
		tmp = y / a;
	else
		tmp = z * (-y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.02e+27], N[(y / a), $MachinePrecision], If[LessEqual[y, 1.3e+89], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+129], N[(N[(y * z), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[y, 2.5e+207], N[(y / a), $MachinePrecision], N[(z * N[((-y) / t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.0199999999999999e27 or 1.55e129 < y < 2.5e207

   1. Initial program 74.7%

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

      1. *-commutative 74.7%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in z around inf 46.3%

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

   if -1.0199999999999999e27 < y < 1.3e89

   1. Initial program 94.6%

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

      1. *-commutative 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in x around inf 72.0%

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

   if 1.3e89 < y < 1.55e129

   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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 84.7%

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

      2. associate-/r/ 87.0%

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

      3. sub-neg 87.0%

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

      4. +-commutative 87.0%

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

      5. *-commutative 87.0%

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

      6. distribute-rgt-neg-in 87.0%

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

      7. fma-define 87.0%

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

   6. Applied egg-rr 87.0%

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

   7. Taylor expanded in a around 0 75.0%

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

   8. Taylor expanded in x around 0 61.6%

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

      1. associate-*r/ 61.6%

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

      2. mul-1-neg 61.6%

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

      3. *-commutative 61.6%

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

      4. distribute-lft-neg-in 61.6%

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

   10. Simplified 61.6%

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

   if 2.5e207 < y

   1. Initial program 64.0%

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

      1. *-commutative 64.0%

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

   3. Simplified 64.0%

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

      1. clear-num 64.1%

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

      2. associate-/r/ 64.0%

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

      3. sub-neg 64.0%

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

      4. +-commutative 64.0%

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

      5. *-commutative 64.0%

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

      6. distribute-rgt-neg-in 64.0%

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

      7. fma-define 64.0%

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

   6. Applied egg-rr 64.0%

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

   7. Taylor expanded in a around 0 51.7%

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

   8. Taylor expanded in x around 0 47.6%

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

      1. mul-1-neg 47.6%

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

      2. *-commutative 47.6%

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

      3. associate-/l* 55.7%

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

      4. distribute-lft-neg-in 55.7%

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

   10. Simplified 55.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+129}:\\ \;\;\;\;\frac{y \cdot z}{-t}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+207}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+52}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} - \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.5e+55)
   (/ (- x (* y z)) t)
   (if (<= t 1.02e+52) (/ (- y (/ x z)) a) (- (/ x t) (/ (* y z) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.5e+55) {
		tmp = (x - (y * z)) / t;
	} else if (t <= 1.02e+52) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x / t) - ((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 (t <= (-2.5d+55)) then
        tmp = (x - (y * z)) / t
    else if (t <= 1.02d+52) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x / t) - ((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 (t <= -2.5e+55) {
		tmp = (x - (y * z)) / t;
	} else if (t <= 1.02e+52) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x / t) - ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -2.5e+55:
		tmp = (x - (y * z)) / t
	elif t <= 1.02e+52:
		tmp = (y - (x / z)) / a
	else:
		tmp = (x / t) - ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.5e+55)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (t <= 1.02e+52)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x / t) - Float64(Float64(y * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -2.5e+55)
		tmp = (x - (y * z)) / t;
	elseif (t <= 1.02e+52)
		tmp = (y - (x / z)) / a;
	else
		tmp = (x / t) - ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.5e+55], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 1.02e+52], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / t), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.50000000000000023e55

   1. Initial program 87.8%

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

      1. *-commutative 87.8%

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

   3. Simplified 87.8%

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

   5. Taylor expanded in t around inf 84.0%

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

   if -2.50000000000000023e55 < t < 1.02000000000000002e52

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in x around 0 86.5%

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

      1. fma-define 86.5%

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

      2. associate-/l* 91.0%

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

      3. cancel-sign-sub-inv 91.0%

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

      4. *-commutative 91.0%

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

      5. +-commutative 91.0%

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

      6. fma-define 91.0%

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

      7. cancel-sign-sub-inv 91.0%

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

      8. *-commutative 91.0%

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

      9. +-commutative 91.0%

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

      10. fma-define 91.0%

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

   7. Simplified 91.0%

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

   8. Taylor expanded in a around inf 70.5%

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

      1. mul-1-neg 70.5%

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

      2. unsub-neg 70.5%

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

   10. Simplified 70.5%

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

   if 1.02000000000000002e52 < t

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

   3. Simplified 80.0%

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

      1. clear-num 79.8%

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

      2. associate-/r/ 79.9%

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

      3. sub-neg 79.9%

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

      4. +-commutative 79.9%

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

      5. *-commutative 79.9%

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

      6. distribute-rgt-neg-in 79.9%

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

      7. fma-define 79.9%

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

   6. Applied egg-rr 79.9%

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

   7. Taylor expanded in a around 0 67.4%

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

      1. associate-*l/ 67.5%

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

      2. *-un-lft-identity 67.5%

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

      3. div-sub 67.6%

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

      4. *-commutative 67.6%

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

   9. Applied egg-rr 67.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+52}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} - \frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{+55} \lor \neg \left(t \leq 6.2 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.25e+55) (not (<= t 6.2e+51)))
   (/ (- x (* y z)) t)
   (/ (- y (/ x z)) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.25e+55) || !(t <= 6.2e+51)) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = (y - (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 ((t <= (-2.25d+55)) .or. (.not. (t <= 6.2d+51))) then
        tmp = (x - (y * z)) / t
    else
        tmp = (y - (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 ((t <= -2.25e+55) || !(t <= 6.2e+51)) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -2.25e+55) or not (t <= 6.2e+51):
		tmp = (x - (y * z)) / t
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.25e+55) || !(t <= 6.2e+51))
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -2.25e+55) || ~((t <= 6.2e+51)))
		tmp = (x - (y * z)) / t;
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.25e+55], N[Not[LessEqual[t, 6.2e+51]], $MachinePrecision]], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.24999999999999999e55 or 6.20000000000000022e51 < t

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in t around inf 75.0%

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

   if -2.24999999999999999e55 < t < 6.20000000000000022e51

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in x around 0 86.5%

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

      1. fma-define 86.5%

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

      2. associate-/l* 91.0%

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

      3. cancel-sign-sub-inv 91.0%

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

      4. *-commutative 91.0%

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

      5. +-commutative 91.0%

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

      6. fma-define 91.0%

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

      7. cancel-sign-sub-inv 91.0%

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

      8. *-commutative 91.0%

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

      9. +-commutative 91.0%

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

      10. fma-define 91.0%

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

   7. Simplified 91.0%

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

   8. Taylor expanded in a around inf 70.5%

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

      1. mul-1-neg 70.5%

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

      2. unsub-neg 70.5%

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

   10. Simplified 70.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{+55} \lor \neg \left(t \leq 6.2 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-17} \lor \neg \left(z \leq 1.06 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1e-17) (not (<= z 1.06e-46))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1e-17) || !(z <= 1.06e-46)) {
		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 <= (-1d-17)) .or. (.not. (z <= 1.06d-46))) 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 <= -1e-17) || !(z <= 1.06e-46)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1e-17) or not (z <= 1.06e-46):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1e-17) || !(z <= 1.06e-46))
		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 <= -1e-17) || ~((z <= 1.06e-46)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1e-17], N[Not[LessEqual[z, 1.06e-46]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.00000000000000007e-17 or 1.06e-46 < z

   1. Initial program 74.6%

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

      1. *-commutative 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in z around inf 48.6%

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

   if -1.00000000000000007e-17 < z < 1.06e-46

   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. Add Preprocessing

   5. Taylor expanded in z around 0 54.3%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-17} \lor \neg \left(z \leq 1.06 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.0% 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 85.3%

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

   1. *-commutative 85.3%

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

3. Simplified 85.3%

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

5. Taylor expanded in z around 0 30.1%

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

6. Final simplification 30.1%

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

Developer target: 97.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024050 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (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))))