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

Percentage Accurate: 93.3% → 97.0%

Time: 9.7s

Alternatives: 11

Speedup: 1.1×

• 24.2% 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: 97.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 - z) / (a / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 93.7

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

4. Applied rewrites 93.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 97.7

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

6. Applied rewrites 97.7%

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

7. Final simplification 97.7%

   \[\leadsto x + \frac{t - z}{\frac{a}{y}} \]
8. Add Preprocessing

Alternative 2: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ t_2 := \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+136}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)) (t_2 (* (/ y a) (- t z))))
   (if (<= t_1 -1e+136) t_2 (if (<= t_1 5e+91) (fma t (/ y a) x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double t_2 = (y / a) * (t - z);
	double tmp;
	if (t_1 <= -1e+136) {
		tmp = t_2;
	} else if (t_1 <= 5e+91) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) * y) / a)
	t_2 = Float64(Float64(y / a) * Float64(t - z))
	tmp = 0.0
	if (t_1 <= -1e+136)
		tmp = t_2;
	elseif (t_1 <= 5e+91)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+136], t$95$2, If[LessEqual[t$95$1, 5e+91], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1.00000000000000006e136 or 5.0000000000000002e91 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 87.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 87.1

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

   4. Applied rewrites 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.8

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 82.2

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

   9. Applied rewrites 82.2%

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

   11. Applied rewrites 91.0%

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

   if -1.00000000000000006e136 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.0000000000000002e91

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.7

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity 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. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 83.3

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

   9. Applied rewrites 83.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -1 \cdot 10^{+136}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ y a) x)))
   (if (<= t -8.5e-44) t_1 (if (<= t 1.26e+40) (- x (/ (* z y) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, (y / a), x);
	double tmp;
	if (t <= -8.5e-44) {
		tmp = t_1;
	} else if (t <= 1.26e+40) {
		tmp = x - ((z * y) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t <= -8.5e-44)
		tmp = t_1;
	elseif (t <= 1.26e+40)
		tmp = Float64(x - Float64(Float64(z * y) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -8.5e-44], t$95$1, If[LessEqual[t, 1.26e+40], N[(x - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.5000000000000002e-44 or 1.2599999999999999e40 < t

   1. Initial program 89.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 89.0

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

   4. Applied rewrites 89.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity 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. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 89.9

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

   9. Applied rewrites 89.9%

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

   if -8.5000000000000002e-44 < t < 1.2599999999999999e40

   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. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.2e+67)
   (/ (* y (- t z)) a)
   (if (<= z 1.05e+193) (fma t (/ y a) x) (* z (/ y (- a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.2e+67) {
		tmp = (y * (t - z)) / a;
	} else if (z <= 1.05e+193) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = z * (y / -a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.2e+67)
		tmp = Float64(Float64(y * Float64(t - z)) / a);
	elseif (z <= 1.05e+193)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = Float64(z * Float64(y / Float64(-a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.2e+67], N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 1.05e+193], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(y / (-a)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.1999999999999994e67

   1. Initial program 91.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -9.1999999999999994e67 < z < 1.05e193

   1. Initial program 95.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 95.0

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

   4. Applied rewrites 95.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.5

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

   6. Applied rewrites 97.5%

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

   7. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity 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. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 87.0

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

   9. Applied rewrites 87.0%

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

   if 1.05e193 < z

   1. Initial program 85.0%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-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. lower-*.f64 N/A

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

      5. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 74.2%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.05e+108)
   (/ (* y (- z)) a)
   (if (<= z 1.05e+193) (fma t (/ y a) x) (* z (/ y (- a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e+108) {
		tmp = (y * -z) / a;
	} else if (z <= 1.05e+193) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = z * (y / -a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.05e+108)
		tmp = Float64(Float64(y * Float64(-z)) / a);
	elseif (z <= 1.05e+193)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = Float64(z * Float64(y / Float64(-a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.05e+108], N[(N[(y * (-z)), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 1.05e+193], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(y / (-a)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05000000000000005e108

   1. Initial program 92.1%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-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. lower-*.f64 N/A

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

      5. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 74.2%

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

   if -1.05000000000000005e108 < z < 1.05e193

   1. Initial program 94.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity 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. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 85.2

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

   9. Applied rewrites 85.2%

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

   if 1.05e193 < z

   1. Initial program 85.0%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-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. lower-*.f64 N/A

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

      5. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 74.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{-a}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ y (- a)))))
   (if (<= z -1.05e+108) t_1 (if (<= z 1.05e+193) (fma t (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (y / -a);
	double tmp;
	if (z <= -1.05e+108) {
		tmp = t_1;
	} else if (z <= 1.05e+193) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(y / Float64(-a)))
	tmp = 0.0
	if (z <= -1.05e+108)
		tmp = t_1;
	elseif (z <= 1.05e+193)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e+108], t$95$1, If[LessEqual[z, 1.05e+193], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000005e108 or 1.05e193 < z

   1. Initial program 89.7%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-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. lower-*.f64 N/A

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

      5. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 74.2%

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

   if -1.05000000000000005e108 < z < 1.05e193

   1. Initial program 94.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity 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. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 85.2

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

   9. Applied rewrites 85.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;z \cdot \frac{y}{-a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;y \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.05e+108) (* y (/ (- z) a)) (fma t (/ y a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e+108) {
		tmp = y * (-z / a);
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.05e+108)
		tmp = Float64(y * Float64(Float64(-z) / a));
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.05e+108], N[(y * N[((-z) / a), $MachinePrecision]), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000005e108

   1. Initial program 92.1%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-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. lower-*.f64 N/A

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

      5. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -1.05000000000000005e108 < z

   1. Initial program 94.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.8

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity 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. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.1

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

   9. Applied rewrites 82.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;y \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% 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 93.7%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
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. Applied rewrites 97.7%

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

Alternative 9: 71.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 93.7

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

4. Applied rewrites 93.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 97.7

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

6. Applied rewrites 97.7%

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

7. Taylor expanded in z around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity 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. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 75.2

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

9. Applied rewrites 75.2%

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

Alternative 10: 68.6% 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 93.7%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
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. lower-fma.f64 N/A

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

   8. lower-/.f64 70.7

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

5. Applied rewrites 70.7%

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

Alternative 11: 33.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return t * (y / 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 = t * (y / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 33.0

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

5. Applied rewrites 33.0%

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

7. Applied rewrites 36.6%

   \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]
8. 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 2024238 
(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)))