Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F

Percentage Accurate: 93.3% → 95.7%

Time: 10.8s

Alternatives: 12

Speedup: 1.1×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(t - z\right) \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -9e-88) (fma (/ (- t z) a) y x) (+ x (/ (* (- t z) y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -9e-88) {
		tmp = fma(((t - z) / a), y, x);
	} else {
		tmp = x + (((t - z) * y) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -9e-88)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	else
		tmp = Float64(x + Float64(Float64(Float64(t - z) * y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -9e-88], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.99999999999999982e-88

   1. Initial program 87.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 96.3%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. sub-neg N/A

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

      8. associate-*l/ N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-frac-neg2 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. frac-2neg N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. sub-neg N/A

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

      18. remove-double-neg N/A

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

      19. /-lowering-/.f64 N/A

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

      20. --lowering--.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -8.99999999999999982e-88 < a

   1. Initial program 97.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(t - z\right) \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+74}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (- t z) (/ y a))))
   (if (<= t_1 -1e+74) t_2 (if (<= t_1 2e+30) (- x (/ (* z y) a)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (t - z) * (y / a);
	double tmp;
	if (t_1 <= -1e+74) {
		tmp = t_2;
	} else if (t_1 <= 2e+30) {
		tmp = x - ((z * y) / a);
	} 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 = (y * (z - t)) / a
    t_2 = (t - z) * (y / a)
    if (t_1 <= (-1d+74)) then
        tmp = t_2
    else if (t_1 <= 2d+30) then
        tmp = x - ((z * y) / a)
    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 = (y * (z - t)) / a;
	double t_2 = (t - z) * (y / a);
	double tmp;
	if (t_1 <= -1e+74) {
		tmp = t_2;
	} else if (t_1 <= 2e+30) {
		tmp = x - ((z * y) / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = (t - z) * (y / a)
	tmp = 0
	if t_1 <= -1e+74:
		tmp = t_2
	elif t_1 <= 2e+30:
		tmp = x - ((z * y) / a)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(Float64(t - z) * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -1e+74)
		tmp = t_2;
	elseif (t_1 <= 2e+30)
		tmp = Float64(x - Float64(Float64(z * y) / a));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	t_2 = (t - z) * (y / a);
	tmp = 0.0;
	if (t_1 <= -1e+74)
		tmp = t_2;
	elseif (t_1 <= 2e+30)
		tmp = x - ((z * y) / a);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+74], t$95$2, If[LessEqual[t$95$1, 2e+30], N[(x - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.99999999999999952e73 or 2e30 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 90.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 90.2

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

   4. Applied egg-rr 90.2%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. neg-sub0 N/A

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

      5. div-sub N/A

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

      6. associate-+l- N/A

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

      7. neg-sub0 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*l/ N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

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

      14. associate-*r/ N/A

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

      15. distribute-rgt-out-- N/A

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

      16. *-lowering-*.f64 N/A

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

      17. /-lowering-/.f64 N/A

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

      18. --lowering--.f64 86.8

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

   7. Simplified 86.8%

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

   if -9.99999999999999952e73 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e30

   1. Initial program 98.4%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 93.7

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

   5. Simplified 93.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1 \cdot 10^{+74}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (- t z) (/ y a))))
   (if (<= t_1 -2e+72) t_2 (if (<= t_1 0.002) (fma (/ y a) t x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (t - z) * (y / a);
	double tmp;
	if (t_1 <= -2e+72) {
		tmp = t_2;
	} else if (t_1 <= 0.002) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(Float64(t - z) * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -2e+72)
		tmp = t_2;
	elseif (t_1 <= 0.002)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+72], t$95$2, If[LessEqual[t$95$1, 0.002], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1.99999999999999989e72 or 2e-3 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 91.2%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 90.4

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

   4. Applied egg-rr 90.4%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. neg-sub0 N/A

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

      5. div-sub N/A

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

      6. associate-+l- N/A

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

      7. neg-sub0 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*l/ N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

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

      14. associate-*r/ N/A

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

      15. distribute-rgt-out-- N/A

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

      16. *-lowering-*.f64 N/A

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

      17. /-lowering-/.f64 N/A

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

      18. --lowering--.f64 86.1

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

   7. Simplified 86.1%

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

   if -1.99999999999999989e72 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e-3

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 97.2%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 85.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := t \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+74}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* t (/ y a))))
   (if (<= t_1 -1e+74) t_2 (if (<= t_1 2e+30) x t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = t * (y / a);
	double tmp;
	if (t_1 <= -1e+74) {
		tmp = t_2;
	} else if (t_1 <= 2e+30) {
		tmp = x;
	} 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 = (y * (z - t)) / a
    t_2 = t * (y / a)
    if (t_1 <= (-1d+74)) then
        tmp = t_2
    else if (t_1 <= 2d+30) then
        tmp = x
    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 = (y * (z - t)) / a;
	double t_2 = t * (y / a);
	double tmp;
	if (t_1 <= -1e+74) {
		tmp = t_2;
	} else if (t_1 <= 2e+30) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = t * (y / a)
	tmp = 0
	if t_1 <= -1e+74:
		tmp = t_2
	elif t_1 <= 2e+30:
		tmp = x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(t * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -1e+74)
		tmp = t_2;
	elseif (t_1 <= 2e+30)
		tmp = x;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	t_2 = t * (y / a);
	tmp = 0.0;
	if (t_1 <= -1e+74)
		tmp = t_2;
	elseif (t_1 <= 2e+30)
		tmp = x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+74], t$95$2, If[LessEqual[t$95$1, 2e+30], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.99999999999999952e73 or 2e30 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 94.9%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 51.5

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

   8. Simplified 51.5%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 53.5

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

   10. Applied egg-rr 53.5%

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

   if -9.99999999999999952e73 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e30

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 77.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1 \cdot 10^{+74}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := y \cdot \frac{t}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+97}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* y (/ t a))))
   (if (<= t_1 -4e+97) t_2 (if (<= t_1 1e+15) x t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = y * (t / a);
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = t_2;
	} else if (t_1 <= 1e+15) {
		tmp = x;
	} 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 = (y * (z - t)) / a
    t_2 = y * (t / a)
    if (t_1 <= (-4d+97)) then
        tmp = t_2
    else if (t_1 <= 1d+15) then
        tmp = x
    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 = (y * (z - t)) / a;
	double t_2 = y * (t / a);
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = t_2;
	} else if (t_1 <= 1e+15) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = y * (t / a)
	tmp = 0
	if t_1 <= -4e+97:
		tmp = t_2
	elif t_1 <= 1e+15:
		tmp = x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(y * Float64(t / a))
	tmp = 0.0
	if (t_1 <= -4e+97)
		tmp = t_2;
	elseif (t_1 <= 1e+15)
		tmp = x;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	t_2 = y * (t / a);
	tmp = 0.0;
	if (t_1 <= -4e+97)
		tmp = t_2;
	elseif (t_1 <= 1e+15)
		tmp = x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+97], t$95$2, If[LessEqual[t$95$1, 1e+15], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4.0000000000000003e97 or 1e15 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 95.0%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 51.5

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

   8. Simplified 51.5%

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

   if -4.0000000000000003e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1e15

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 77.8%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+37}:\\ \;\;\;\;\frac{z \cdot y}{-a}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e+37)
   (/ (* z y) (- a))
   (if (<= z 1.25e+164) (fma (/ y a) t x) (- (* z (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e+37) {
		tmp = (z * y) / -a;
	} else if (z <= 1.25e+164) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = -(z * (y / a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e+37)
		tmp = Float64(Float64(z * y) / Float64(-a));
	elseif (z <= 1.25e+164)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(-Float64(z * Float64(y / a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e+37], N[(N[(z * y), $MachinePrecision] / (-a)), $MachinePrecision], If[LessEqual[z, 1.25e+164], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], (-N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if z < -2.69999999999999986e37

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 94.6%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. neg-lowering-neg.f64 68.2

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

   8. Simplified 68.2%

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

   if -2.69999999999999986e37 < z < 1.24999999999999987e164

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 96.4%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 85.0%

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

   if 1.24999999999999987e164 < z

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 96.3%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. neg-lowering-neg.f64 79.3

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

   8. Simplified 79.3%

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

      1. associate-/l* N/A

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

      2. distribute-frac-neg N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 82.5

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

   10. Applied egg-rr 82.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+37}:\\ \;\;\;\;\frac{z \cdot y}{-a}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+37}:\\ \;\;\;\;-y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 3.85 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e+37)
   (- (* y (/ z a)))
   (if (<= z 3.85e+157) (fma (/ y a) t x) (- (* z (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e+37) {
		tmp = -(y * (z / a));
	} else if (z <= 3.85e+157) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = -(z * (y / a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e+37)
		tmp = Float64(-Float64(y * Float64(z / a)));
	elseif (z <= 3.85e+157)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(-Float64(z * Float64(y / a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e+37], (-N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[z, 3.85e+157], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], (-N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if z < -2.69999999999999986e37

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 63.8

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

   5. Simplified 63.8%

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

   if -2.69999999999999986e37 < z < 3.8500000000000002e157

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 96.4%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 85.0%

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

   if 3.8500000000000002e157 < z

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 96.3%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. neg-lowering-neg.f64 79.3

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

   8. Simplified 79.3%

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

      1. associate-/l* N/A

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

      2. distribute-frac-neg N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 82.5

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

   10. Applied egg-rr 82.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+37}:\\ \;\;\;\;-y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 3.85 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -y \cdot \frac{z}{a}\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.7 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* y (/ z a)))))
   (if (<= z -2.7e+37) t_1 (if (<= z 8.7e+158) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -(y * (z / a));
	double tmp;
	if (z <= -2.7e+37) {
		tmp = t_1;
	} else if (z <= 8.7e+158) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(-Float64(y * Float64(z / a)))
	tmp = 0.0
	if (z <= -2.7e+37)
		tmp = t_1;
	elseif (z <= 8.7e+158)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = (-N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[z, -2.7e+37], t$95$1, If[LessEqual[z, 8.7e+158], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.69999999999999986e37 or 8.70000000000000024e158 < z

   1. Initial program 95.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 68.5

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

   5. Simplified 68.5%

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

   if -2.69999999999999986e37 < z < 8.70000000000000024e158

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Simplified 96.4%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 85.0%

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

Alternative 9: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{a}, t - z, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in x around 0

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

   1. associate-*l/ N/A

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

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

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

   3. associate-*l/ N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. associate-+l- N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate-+r+ N/A

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

5. Simplified 96.1%

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

Alternative 10: 70.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{a}, t, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in x around 0

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

   1. associate-*l/ N/A

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

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

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

   3. associate-*l/ N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. associate-+l- N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate-+r+ N/A

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

5. Simplified 96.1%

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

6. Taylor expanded in t around inf

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

8. Simplified 71.6%

   \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]
9. Add Preprocessing

Alternative 11: 67.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \frac{t}{a}, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in z around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. /-lowering-/.f64 70.5

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

5. Simplified 70.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{a}, x\right)} \]
6. Add Preprocessing

Alternative 12: 39.3% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 x)

C

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

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
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in x around inf

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

5. Simplified 42.5%

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

Developer Target 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{t\_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (- x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (- x (/ (* y (- z t)) a))
       (- x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / 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 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x - (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x - ((y * (z - t)) / a)
    else
        tmp = x - (y / 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 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x - (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x - ((y * (z - t)) / a)
	else:
		tmp = x - (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x - Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x - Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x - Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x - (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x - ((y * (z - t)) / a);
	else
		tmp = x - (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x - N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -430450648655599/4000000000000000000000000) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t)))))))

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