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

Percentage Accurate: 93.0% → 97.3%

Time: 2.3s

Alternatives: 8

Speedup: 1.1×

• 74.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. mult-flip-rev N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 97.3%

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

3. Applied rewrites 97.3%

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

Alternative 2: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{if}\;t \leq -4.05 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) t x)))
   (if (<= t -4.05e+46) t_1 (if (<= t 3.5e-9) (- x (/ (* y z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), t, x);
	double tmp;
	if (t <= -4.05e+46) {
		tmp = t_1;
	} else if (t <= 3.5e-9) {
		tmp = x - ((y * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), t, x)
	tmp = 0.0
	if (t <= -4.05e+46)
		tmp = t_1;
	elseif (t <= 3.5e-9)
		tmp = Float64(x - Float64(Float64(y * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[t, -4.05e+46], t$95$1, If[LessEqual[t, 3.5e-9], N[(x - N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.05000000000000024e46 or 3.4999999999999999e-9 < t

   1. Initial program 93.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.3%

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

   3. Applied rewrites 97.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 70.9%

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

   if -4.05000000000000024e46 < t < 3.4999999999999999e-9

   1. Initial program 93.0%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 68.2%

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

   4. Applied rewrites 68.2%

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

Alternative 3: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+239}:\\ \;\;\;\;\frac{t - z}{a} \cdot y\\ \mathbf{elif}\;t\_1 \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 -5e+239)
     (* (/ (- t z) a) y)
     (if (<= t_1 200000000.0) (fma (/ y a) t x) (* (/ y a) (- t z))))))

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+239) {
		tmp = ((t - z) / a) * y;
	} else if (t_1 <= 200000000.0) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = (y / a) * (t - z);
	}
	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+239)
		tmp = Float64(Float64(Float64(t - z) / a) * y);
	elseif (t_1 <= 200000000.0)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(Float64(y / a) * Float64(t - z));
	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[LessEqual[t$95$1, -5e+239], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 200000000.0], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.00000000000000007e239

   1. Initial program 93.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 56.2%

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

   4. Applied rewrites 56.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.2%

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-div N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 57.7%

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

   6. Applied rewrites 57.7%

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

   if -5.00000000000000007e239 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e8

   1. Initial program 93.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.3%

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

   3. Applied rewrites 97.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 70.9%

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

   if 2e8 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 56.2%

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

   4. Applied rewrites 56.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sub-div N/A

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

      6. lift--.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 60.2%

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

   6. Applied rewrites 60.2%

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

Alternative 4: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+239}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

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+239) t_2 (if (<= t_1 200000000.0) (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+239) {
		tmp = t_2;
	} else if (t_1 <= 200000000.0) {
		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+239)
		tmp = t_2;
	elseif (t_1 <= 200000000.0)
		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+239], t$95$2, If[LessEqual[t$95$1, 200000000.0], 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) < -5.00000000000000007e239 or 2e8 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 56.2%

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

   4. Applied rewrites 56.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sub-div N/A

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

      6. lift--.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 60.2%

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

   6. Applied rewrites 60.2%

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

   if -5.00000000000000007e239 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e8

   1. Initial program 93.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.3%

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

   3. Applied rewrites 97.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 70.9%

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

Alternative 5: 70.9% accurate, 1.3× speedup

Math

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

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 93.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. mult-flip-rev N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 97.3%

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

3. Applied rewrites 97.3%

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

4. Taylor expanded in z around 0

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

6. Applied rewrites 70.9%

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

Alternative 6: 59.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{y}{a} \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+119}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ y a) t)))
   (if (<= t_1 -2e+119) t_2 (if (<= t_1 2e+44) 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 <= -2e+119) {
		tmp = t_2;
	} else if (t_1 <= 2e+44) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    t_2 = (y / a) * t
    if (t_1 <= (-2d+119)) then
        tmp = t_2
    else if (t_1 <= 2d+44) 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 <= -2e+119) {
		tmp = t_2;
	} else if (t_1 <= 2e+44) {
		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 <= -2e+119:
		tmp = t_2
	elif t_1 <= 2e+44:
		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 <= -2e+119)
		tmp = t_2;
	elseif (t_1 <= 2e+44)
		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 <= -2e+119)
		tmp = t_2;
	elseif (t_1 <= 2e+44)
		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, -2e+119], t$95$2, If[LessEqual[t$95$1, 2e+44], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1.99999999999999989e119 or 2.0000000000000002e44 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.0%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.5%

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

   4. Applied rewrites 31.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.4%

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

   6. Applied rewrites 34.4%

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

   if -1.99999999999999989e119 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2.0000000000000002e44

   1. Initial program 93.0%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.5%

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

   4. Applied rewrites 31.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.4%

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

   6. Applied rewrites 34.4%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 39.0%

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

Alternative 7: 56.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{t}{a} \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+239}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+167}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ t a) y)))
   (if (<= t_1 -5e+239) t_2 (if (<= t_1 1e+167) x t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (t / a) * y;
	double tmp;
	if (t_1 <= -5e+239) {
		tmp = t_2;
	} else if (t_1 <= 1e+167) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    t_2 = (t / a) * y
    if (t_1 <= (-5d+239)) then
        tmp = t_2
    else if (t_1 <= 1d+167) then
        tmp = x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (t / a) * y;
	double tmp;
	if (t_1 <= -5e+239) {
		tmp = t_2;
	} else if (t_1 <= 1e+167) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = (t / a) * y
	tmp = 0
	if t_1 <= -5e+239:
		tmp = t_2
	elif t_1 <= 1e+167:
		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(t / a) * y)
	tmp = 0.0
	if (t_1 <= -5e+239)
		tmp = t_2;
	elseif (t_1 <= 1e+167)
		tmp = x;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	t_2 = (t / a) * y;
	tmp = 0.0;
	if (t_1 <= -5e+239)
		tmp = t_2;
	elseif (t_1 <= 1e+167)
		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[(t / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+239], t$95$2, If[LessEqual[t$95$1, 1e+167], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.00000000000000007e239 or 1e167 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.0%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.5%

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

   4. Applied rewrites 31.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 32.0%

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

   6. Applied rewrites 32.0%

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

   if -5.00000000000000007e239 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1e167

   1. Initial program 93.0%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.5%

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

   4. Applied rewrites 31.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.4%

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

   6. Applied rewrites 34.4%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 39.0%

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

Alternative 8: 39.0% accurate, 12.6× speedup

Math

\[x \]

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

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
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 93.0%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 31.5%

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

4. Applied rewrites 31.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 34.4%

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

6. Applied rewrites 34.4%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 39.0%

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

Reproduce

herbie shell --seed 2025188 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64
  (- x (/ (* y (- z t)) a)))