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

Percentage Accurate: 92.9% → 97.7%

Time: 9.5s

Alternatives: 12

Speedup: 1.1×

• 24.8% 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: 92.9% 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: 97.7% 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 91.7%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
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 97.1%

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

Alternative 2: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ y a) (- t z))))
   (if (<= t_1 -5e-34) t_2 (if (<= t_1 5e+82) (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 = (y / a) * (t - z);
	double tmp;
	if (t_1 <= -5e-34) {
		tmp = t_2;
	} else if (t_1 <= 5e+82) {
		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(y / a) * Float64(t - z))
	tmp = 0.0
	if (t_1 <= -5e-34)
		tmp = t_2;
	elseif (t_1 <= 5e+82)
		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[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-34], t$95$2, If[LessEqual[t$95$1, 5e+82], 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) < -5.0000000000000003e-34 or 5.00000000000000015e82 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.8%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. /-lowering-/.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. *-lowering-*.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. --lowering--.f64 79.9

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

   5. Simplified 79.9%

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

      1. associate-*l/ N/A

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

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

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

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

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

      4. --lowering--.f64 87.0

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

   7. Applied egg-rr 87.0%

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

   if -5.0000000000000003e-34 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.00000000000000015e82

   1. Initial program 96.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 96.6%

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

   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 3: 60.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{y}{a} \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+52}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ y a) t)))
   (if (<= t_1 -1e+52) t_2 (if (<= t_1 5e+82) 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 / a) * t;
	double tmp;
	if (t_1 <= -1e+52) {
		tmp = t_2;
	} else if (t_1 <= 5e+82) {
		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 / a) * t
    if (t_1 <= (-1d+52)) then
        tmp = t_2
    else if (t_1 <= 5d+82) 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 / a) * t;
	double tmp;
	if (t_1 <= -1e+52) {
		tmp = t_2;
	} else if (t_1 <= 5e+82) {
		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 / a) * t
	tmp = 0
	if t_1 <= -1e+52:
		tmp = t_2
	elif t_1 <= 5e+82:
		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(Float64(y / a) * t)
	tmp = 0.0
	if (t_1 <= -1e+52)
		tmp = t_2;
	elseif (t_1 <= 5e+82)
		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 / a) * t;
	tmp = 0.0;
	if (t_1 <= -1e+52)
		tmp = t_2;
	elseif (t_1 <= 5e+82)
		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[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+52], t$95$2, If[LessEqual[t$95$1, 5e+82], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.9999999999999999e51 or 5.00000000000000015e82 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.2%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 49.1

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

   5. Simplified 49.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 52.5

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

   7. Applied egg-rr 52.5%

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

   if -9.9999999999999999e51 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.00000000000000015e82

   1. Initial program 96.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 67.3%

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

Alternative 4: 57.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := y \cdot \frac{t}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+52}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+234}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* y (/ t a))))
   (if (<= t_1 -1e+52) t_2 (if (<= t_1 1e+234) 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 <= -1e+52) {
		tmp = t_2;
	} else if (t_1 <= 1e+234) {
		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 <= (-1d+52)) then
        tmp = t_2
    else if (t_1 <= 1d+234) 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 <= -1e+52) {
		tmp = t_2;
	} else if (t_1 <= 1e+234) {
		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 <= -1e+52:
		tmp = t_2
	elif t_1 <= 1e+234:
		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 <= -1e+52)
		tmp = t_2;
	elseif (t_1 <= 1e+234)
		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 <= -1e+52)
		tmp = t_2;
	elseif (t_1 <= 1e+234)
		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, -1e+52], t$95$2, If[LessEqual[t$95$1, 1e+234], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.9999999999999999e51 or 1.00000000000000002e234 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 86.5%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 50.9

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

   5. Simplified 50.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 50.3

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

   7. Applied egg-rr 50.3%

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

   if -9.9999999999999999e51 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.00000000000000002e234

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.4%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 10^{+234}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+302}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 -1e+302)
     (* (/ y a) (- t z))
     (if (<= t_1 1e+110) (fma (/ y a) t x) (* y (/ (- t z) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -1e+302) {
		tmp = (y / a) * (t - z);
	} else if (t_1 <= 1e+110) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = y * ((t - z) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= -1e+302)
		tmp = Float64(Float64(y / a) * Float64(t - z));
	elseif (t_1 <= 1e+110)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(y * Float64(Float64(t - z) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+302], N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+110], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -1.0000000000000001e302

   1. Initial program 76.3%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. /-lowering-/.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. *-lowering-*.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. --lowering--.f64 76.3

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

   5. Simplified 76.3%

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

      1. associate-*l/ N/A

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

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

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

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

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

      4. --lowering--.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -1.0000000000000001e302 < (*.f64 y (-.f64 z t)) < 1e110

   1. Initial program 97.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 95.7%

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

   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 80.7%

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

   if 1e110 < (*.f64 y (-.f64 z t))

   1. Initial program 88.6%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. /-lowering-/.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. *-lowering-*.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. --lowering--.f64 83.9

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

   5. Simplified 83.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. --lowering--.f64 90.4

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -1 \cdot 10^{+302}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, -z, x\right)\\ \mathbf{if}\;z \leq -9 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) (- z) x)))
   (if (<= z -9e+39) t_1 (if (<= z 9.5e+71) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), -z, x);
	double tmp;
	if (z <= -9e+39) {
		tmp = t_1;
	} else if (z <= 9.5e+71) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), Float64(-z), x)
	tmp = 0.0
	if (z <= -9e+39)
		tmp = t_1;
	elseif (z <= 9.5e+71)
		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[(N[(y / a), $MachinePrecision] * (-z) + x), $MachinePrecision]}, If[LessEqual[z, -9e+39], t$95$1, If[LessEqual[z, 9.5e+71], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999991e39 or 9.50000000000000015e71 < z

   1. Initial program 87.3%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 96.7%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 84.9

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

   8. Simplified 84.9%

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

   if -8.99999999999999991e39 < z < 9.50000000000000015e71

   1. Initial program 94.8%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 97.4%

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

   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 91.3%

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

Alternative 7: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot \frac{z}{a}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y (/ z a)))))
   (if (<= z -2.8e+40) t_1 (if (<= z 1e+66) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * (z / a));
	double tmp;
	if (z <= -2.8e+40) {
		tmp = t_1;
	} else if (z <= 1e+66) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * Float64(z / a)))
	tmp = 0.0
	if (z <= -2.8e+40)
		tmp = t_1;
	elseif (z <= 1e+66)
		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[(x - N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e+40], t$95$1, If[LessEqual[z, 1e+66], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e40 or 9.99999999999999945e65 < z

   1. Initial program 87.3%

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

   3. Taylor expanded in t around 0

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

      1. --lowering--.f64 N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 81.7

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

   5. Simplified 81.7%

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

   if -2.8000000000000001e40 < z < 9.99999999999999945e65

   1. Initial program 94.8%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 97.4%

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

   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 91.3%

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

Alternative 8: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+167}:\\ \;\;\;\;-y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.15e+167)
   (- (* y (/ z a)))
   (if (<= z 4.8e+155) (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 <= -1.15e+167) {
		tmp = -(y * (z / a));
	} else if (z <= 4.8e+155) {
		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 <= -1.15e+167)
		tmp = Float64(-Float64(y * Float64(z / a)));
	elseif (z <= 4.8e+155)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(z * Float64(Float64(-y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.15e+167], (-N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[z, 4.8e+155], 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 < -1.14999999999999994e167

   1. Initial program 87.6%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 78.1

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

   5. Simplified 78.1%

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

   if -1.14999999999999994e167 < z < 4.80000000000000042e155

   1. Initial program 93.6%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 97.4%

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

   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 83.1%

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

   if 4.80000000000000042e155 < z

   1. Initial program 83.4%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 70.5

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

   5. Simplified 70.5%

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

      1. distribute-lft-neg-in N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*l/ N/A

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

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

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

      9. associate-*l/ N/A

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

      10. neg-mul-1 N/A

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

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

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

      12. neg-lowering-neg.f64 73.8

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

   7. Applied egg-rr 73.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+167}:\\ \;\;\;\;-y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -y \cdot \frac{z}{a}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* y (/ z a)))))
   (if (<= z -6.5e+171) t_1 (if (<= z 7.2e+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 <= -6.5e+171) {
		tmp = t_1;
	} else if (z <= 7.2e+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 <= -6.5e+171)
		tmp = t_1;
	elseif (z <= 7.2e+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, -6.5e+171], t$95$1, If[LessEqual[z, 7.2e+158], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5e171 or 7.19999999999999976e158 < z

   1. Initial program 85.5%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 74.4

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

   5. Simplified 74.4%

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

   if -6.5e171 < z < 7.19999999999999976e158

   1. Initial program 93.6%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 97.4%

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

   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 83.1%

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

Alternative 10: 71.6% 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 91.7%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
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 97.1%

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

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 70.2%

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

Alternative 11: 68.4% 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 91.7%

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

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

   8. /-lowering-/.f64 67.0

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

5. Simplified 67.0%

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

Alternative 12: 39.6% 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 91.7%

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

3. Taylor expanded in x around inf

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

5. Simplified 35.5%

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

Developer Target 1: 99.2% 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 2024204 
(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)))