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

Percentage Accurate: 93.0% → 98.8%

Time: 4.4s

Alternatives: 11

Speedup: 1.1×

• 24.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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: 93.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, -y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* y (- z t)) a))))
   (if (<= t_1 -5e+295)
     (fma (- t z) (/ y a) x)
     (if (<= t_1 5e+280) t_1 (fma (/ (- z t) a) (- y) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y * (z - t)) / a);
	double tmp;
	if (t_1 <= -5e+295) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 5e+280) {
		tmp = t_1;
	} else {
		tmp = fma(((z - t) / a), -y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y * Float64(z - t)) / a))
	tmp = 0.0
	if (t_1 <= -5e+295)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 5e+280)
		tmp = t_1;
	else
		tmp = fma(Float64(Float64(z - t) / a), Float64(-y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+295], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+280], t$95$1, N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * (-y) + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < -4.99999999999999991e295

   1. Initial program 82.7%

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

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   if -4.99999999999999991e295 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 5.0000000000000002e280

   1. Initial program 99.8%

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

   if 5.0000000000000002e280 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 75.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+295} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+302}\right):\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* y (- z t)) a))))
   (if (or (<= t_1 -5e+295) (not (<= t_1 5e+302)))
     (fma (- t z) (/ y a) x)
     t_1)))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y * (z - t)) / a);
	double tmp;
	if ((t_1 <= -5e+295) || !(t_1 <= 5e+302)) {
		tmp = fma((t - z), (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y * Float64(z - t)) / a))
	tmp = 0.0
	if ((t_1 <= -5e+295) || !(t_1 <= 5e+302))
		tmp = fma(Float64(t - z), Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+295], N[Not[LessEqual[t$95$1, 5e+302]], $MachinePrecision]], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < -4.99999999999999991e295 or 5e302 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   if -4.99999999999999991e295 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 5e302

   1. Initial program 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{y \cdot \left(z - t\right)}{a} \leq -5 \cdot 10^{+295} \lor \neg \left(x - \frac{y \cdot \left(z - t\right)}{a} \leq 5 \cdot 10^{+302}\right):\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-23} \lor \neg \left(t\_1 \leq 10^{+24}\right):\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (or (<= t_1 -5e-23) (not (<= t_1 1e+24)))
     (* (- t z) (/ y a))
     (fma (/ t a) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if ((t_1 <= -5e-23) || !(t_1 <= 1e+24)) {
		tmp = (t - z) * (y / a);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if ((t_1 <= -5e-23) || !(t_1 <= 1e+24))
		tmp = Float64(Float64(t - z) * Float64(y / a));
	else
		tmp = fma(Float64(t / a), y, x);
	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]}, If[Or[LessEqual[t$95$1, -5e-23], N[Not[LessEqual[t$95$1, 1e+24]], $MachinePrecision]], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.0000000000000002e-23 or 9.9999999999999998e23 < (/.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 x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. *-lft-identity N/A

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

      12. fp-cancel-sign-sub-inv N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

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

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

      16. *-commutative N/A

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

      17. mul-1-neg N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 N/A

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

      24. lower-/.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if -5.0000000000000002e-23 < (/.f64 (*.f64 y (-.f64 z t)) a) < 9.9999999999999998e23

   1. Initial program 99.9%

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

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. /-rgt-identity N/A

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

      8. /-rgt-identity N/A

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

      9. associate-*l/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 89.4

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

   5. Applied rewrites 89.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -5 \cdot 10^{-23} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 10^{+24}\right):\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-8} \lor \neg \left(z \leq 7.9 \cdot 10^{+62}\right):\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.06e-8) (not (<= z 7.9e+62)))
   (- x (* z (/ y a)))
   (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.06e-8) || !(z <= 7.9e+62)) {
		tmp = x - (z * (y / a));
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.06e-8) || !(z <= 7.9e+62))
		tmp = Float64(x - Float64(z * Float64(y / a)));
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.06e-8], N[Not[LessEqual[z, 7.9e+62]], $MachinePrecision]], N[(x - N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.06000000000000006e-8 or 7.8999999999999997e62 < z

   1. Initial program 89.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

   7. Applied rewrites 83.3%

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

   if -1.06000000000000006e-8 < z < 7.8999999999999997e62

   1. Initial program 95.3%

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

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-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

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 89.3

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

   8. Applied rewrites 89.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-8} \lor \neg \left(z \leq 7.9 \cdot 10^{+62}\right):\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-8}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.06e-8)
   (- x (* z (/ y a)))
   (if (<= z 3.25e+63) (fma (/ y a) t x) (- x (* (/ z a) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.06e-8) {
		tmp = x - (z * (y / a));
	} else if (z <= 3.25e+63) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = x - ((z / a) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.06e-8)
		tmp = Float64(x - Float64(z * Float64(y / a)));
	elseif (z <= 3.25e+63)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(x - Float64(Float64(z / a) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.06e-8], N[(x - N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.25e+63], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(x - N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.06000000000000006e-8

   1. Initial program 89.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 75.2%

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

   if -1.06000000000000006e-8 < z < 3.24999999999999996e63

   1. Initial program 95.3%

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

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-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

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 89.3

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

   8. Applied rewrites 89.3%

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

   if 3.24999999999999996e63 < z

   1. Initial program 89.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 95.8

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

   5. Applied rewrites 95.8%

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

Alternative 6: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+110} \lor \neg \left(z \leq 5.5 \cdot 10^{+163}\right):\\ \;\;\;\;\frac{-z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.75e+110) (not (<= z 5.5e+163)))
   (* (/ (- z) a) y)
   (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.75e+110) || !(z <= 5.5e+163)) {
		tmp = (-z / a) * y;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.75e+110) || !(z <= 5.5e+163))
		tmp = Float64(Float64(Float64(-z) / a) * y);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.75e+110], N[Not[LessEqual[z, 5.5e+163]], $MachinePrecision]], N[(N[((-z) / a), $MachinePrecision] * y), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.74999999999999998e110 or 5.50000000000000014e163 < z

   1. Initial program 87.8%

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

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if -2.74999999999999998e110 < z < 5.50000000000000014e163

   1. Initial program 94.7%

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

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-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

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 84.6

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

   8. Applied rewrites 84.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+110} \lor \neg \left(z \leq 5.5 \cdot 10^{+163}\right):\\ \;\;\;\;\frac{-z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+135} \lor \neg \left(z \leq 5.5 \cdot 10^{+163}\right):\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -9.8e+135) (not (<= z 5.5e+163)))
   (* (- z) (/ y a))
   (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -9.8e+135) || !(z <= 5.5e+163)) {
		tmp = -z * (y / a);
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -9.8e+135) || !(z <= 5.5e+163))
		tmp = Float64(Float64(-z) * Float64(y / a));
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -9.8e+135], N[Not[LessEqual[z, 5.5e+163]], $MachinePrecision]], N[((-z) * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.8000000000000002e135 or 5.50000000000000014e163 < z

   1. Initial program 88.3%

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

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 70.1%

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

   if -9.8000000000000002e135 < z < 5.50000000000000014e163

   1. Initial program 94.3%

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

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-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

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 84.0

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

   8. Applied rewrites 84.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+135} \lor \neg \left(z \leq 5.5 \cdot 10^{+163}\right):\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

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

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

   9. lower-fma.f64 N/A

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

5. Applied rewrites 95.6%

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

Alternative 9: 71.5% 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 92.8%

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

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

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

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

   9. lower-fma.f64 N/A

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

5. Applied rewrites 95.6%

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

6. Taylor expanded in z around 0

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

   1. fp-cancel-sub-sign-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

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. associate-*l/ N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 71.7

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

8. Applied rewrites 71.7%

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

Alternative 10: 68.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. +-commutative N/A

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

   6. div-add N/A

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

   7. /-rgt-identity N/A

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

   8. /-rgt-identity N/A

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

   9. associate-*l/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 69.7

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

5. Applied rewrites 69.7%

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

Alternative 11: 34.7% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 92.8%

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

3. Taylor expanded in t around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 35.6

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

5. Applied rewrites 35.6%

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

7. Applied rewrites 37.6%

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 2025017 
(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)))