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

Percentage Accurate: 93.3% → 96.1%

Time: 2.7s

Alternatives: 11

Speedup: 1.0×

• 74.6% 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: 93.3% 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: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x - \frac{y \cdot \left(z - t\right)}{a} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= (- x (/ (* y (- z t)) a)) 2e-60)
  (fma (/ (- t z) a) y x)
  (fma (/ y a) (- t z) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x - ((y * (z - t)) / a)) <= 2e-60) {
		tmp = fma(((t - z) / a), y, x);
	} else {
		tmp = fma((y / a), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x - Float64(Float64(y * Float64(z - t)) / a)) <= 2e-60)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	else
		tmp = fma(Float64(y / a), Float64(t - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], 2e-60], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 1.9999999999999999e-60

   1. Initial program 93.3%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 93.5%

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

   3. Applied rewrites 93.5%

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

   if 1.9999999999999999e-60 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 93.3%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 97.5%

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

   3. Applied rewrites 97.5%

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

Alternative 2: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. distribute-neg-frac N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 93.5%

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

3. Applied rewrites 93.5%

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

Alternative 3: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-46}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (/ t a) y x)))
  (if (<= t -1e+87) t_1 (if (<= t 9.5e-46) (- x (/ (* y z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / a), y, x);
	double tmp;
	if (t <= -1e+87) {
		tmp = t_1;
	} else if (t <= 9.5e-46) {
		tmp = x - ((y * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / a), y, x)
	tmp = 0.0
	if (t <= -1e+87)
		tmp = t_1;
	elseif (t <= 9.5e-46)
		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[(t / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -1e+87], t$95$1, If[LessEqual[t, 9.5e-46], N[(x - N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.9999999999999996e86 or 9.4999999999999999e-46 < t

   1. Initial program 93.3%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 93.5%

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 67.9%

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

   if -9.9999999999999996e86 < t < 9.4999999999999999e-46

   1. Initial program 93.3%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 68.7%

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

   4. Applied rewrites 68.7%

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

Alternative 4: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - z}{a} \cdot y\\ t_2 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -25000000000:\\ \;\;\;\;\frac{\left(t - z\right) \cdot y}{a}\\ \mathbf{elif}\;t\_2 \leq 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (* (/ (- t z) a) y)) (t_2 (/ (* y (- z t)) a)))
  (if (<= t_2 (- INFINITY))
    t_1
    (if (<= t_2 -25000000000.0)
      (/ (* (- t z) y) a)
      (if (<= t_2 1e+100) (fma (/ t a) y x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((t - z) / a) * y;
	double t_2 = (y * (z - t)) / a;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -25000000000.0) {
		tmp = ((t - z) * y) / a;
	} else if (t_2 <= 1e+100) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(t - z) / a) * y)
	t_2 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -25000000000.0)
		tmp = Float64(Float64(Float64(t - z) * y) / a);
	elseif (t_2 <= 1e+100)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -25000000000.0], N[(N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$2, 1e+100], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -inf.0 or 1e100 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.3%

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

   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 55.0%

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

   4. Applied rewrites 55.0%

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)} \]
   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 55.0%

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

      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 56.8%

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

   6. Applied rewrites 56.8%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a) < -2.5e10

   1. Initial program 93.3%

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

   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 55.0%

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

   4. Applied rewrites 55.0%

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)} \]
   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. sub-negate-rev N/A

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

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

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-div N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-out N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. lower-*.f64 56.7%

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

   6. Applied rewrites 56.7%

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

   if -2.5e10 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1e100

   1. Initial program 93.3%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 93.5%

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 67.9%

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

Alternative 5: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -25000000000:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - z}{a} \cdot y\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (/ (* y (- z t)) a)))
  (if (<= t_1 -25000000000.0)
    (* (/ y a) (- t z))
    (if (<= t_1 1e+100) (fma (/ t a) y x) (* (/ (- t z) a) y)))))

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 <= -25000000000.0) {
		tmp = (y / a) * (t - z);
	} else if (t_1 <= 1e+100) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = ((t - z) / a) * y;
	}
	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 <= -25000000000.0)
		tmp = Float64(Float64(y / a) * Float64(t - z));
	elseif (t_1 <= 1e+100)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = Float64(Float64(Float64(t - z) / a) * y);
	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, -25000000000.0], N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+100], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -2.5e10

   1. Initial program 93.3%

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

   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 55.0%

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

   4. Applied rewrites 55.0%

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)} \]
   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 60.1%

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

   6. Applied rewrites 60.1%

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

   if -2.5e10 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1e100

   1. Initial program 93.3%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 93.5%

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 67.9%

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

   if 1e100 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.3%

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

   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 55.0%

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

   4. Applied rewrites 55.0%

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)} \]
   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 55.0%

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

      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 56.8%

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

   6. Applied rewrites 56.8%

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

Alternative 6: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{t - z}{a} \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+135}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ (- t z) a) y)))
  (if (<= t_1 -2e+135)
    t_2
    (if (<= t_1 1e+100) (fma (/ t a) y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = ((t - z) / a) * y;
	double tmp;
	if (t_1 <= -2e+135) {
		tmp = t_2;
	} else if (t_1 <= 1e+100) {
		tmp = fma((t / a), y, 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(Float64(t - z) / a) * y)
	tmp = 0.0
	if (t_1 <= -2e+135)
		tmp = t_2;
	elseif (t_1 <= 1e+100)
		tmp = fma(Float64(t / a), y, 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[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+135], t$95$2, If[LessEqual[t$95$1, 1e+100], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1.9999999999999999e135 or 1e100 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.3%

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

   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 55.0%

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

   4. Applied rewrites 55.0%

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)} \]
   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 55.0%

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

      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 56.8%

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

   6. Applied rewrites 56.8%

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

   if -1.9999999999999999e135 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1e100

   1. Initial program 93.3%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 93.5%

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 67.9%

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

Alternative 7: 71.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= (* y (- z t)) -5e-89) (fma (/ y a) t x) (fma (/ t a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * (z - t)) <= -5e-89) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(y * Float64(z - t)) <= -5e-89)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision], -5e-89], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -4.9999999999999997e-89

   1. Initial program 93.3%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 97.5%

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

   3. Applied rewrites 97.5%

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

   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 71.2%

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

   if -4.9999999999999997e-89 < (*.f64 y (-.f64 z t))

   1. Initial program 93.3%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 93.5%

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 67.9%

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

Alternative 8: 69.6% 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 93.3%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
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 97.5%

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

3. Applied rewrites 97.5%

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

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 71.2%

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

Alternative 9: 32.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5 \cdot 10^{-89}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= (* y (- z t)) -5e-89) (/ t (/ a y)) (* (/ t a) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * (z - t)) <= -5e-89) {
		tmp = t / (a / y);
	} else {
		tmp = (t / a) * y;
	}
	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) :: tmp
    if ((y * (z - t)) <= (-5d-89)) then
        tmp = t / (a / y)
    else
        tmp = (t / a) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * (z - t)) <= -5e-89) {
		tmp = t / (a / y);
	} else {
		tmp = (t / a) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y * (z - t)) <= -5e-89:
		tmp = t / (a / y)
	else:
		tmp = (t / a) * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(y * Float64(z - t)) <= -5e-89)
		tmp = Float64(t / Float64(a / y));
	else
		tmp = Float64(Float64(t / a) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y * (z - t)) <= -5e-89)
		tmp = t / (a / y);
	else
		tmp = (t / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision], -5e-89], N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -4.9999999999999997e-89

   1. Initial program 93.3%

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

   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 31.3%

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

   4. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\frac{t \cdot y}{a}} \]
   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 31.3%

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

   6. Applied rewrites 31.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-flip-rev N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 34.0%

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

   8. Applied rewrites 34.0%

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

   if -4.9999999999999997e-89 < (*.f64 y (-.f64 z t))

   1. Initial program 93.3%

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

   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 31.3%

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

   4. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\frac{t \cdot y}{a}} \]
   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 31.3%

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

   6. Applied rewrites 31.3%

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

Alternative 10: 32.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5 \cdot 10^{-89}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= (* y (- z t)) -5e-89) (* (/ y a) t) (* (/ t a) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * (z - t)) <= -5e-89) {
		tmp = (y / a) * t;
	} else {
		tmp = (t / a) * y;
	}
	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) :: tmp
    if ((y * (z - t)) <= (-5d-89)) then
        tmp = (y / a) * t
    else
        tmp = (t / a) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * (z - t)) <= -5e-89) {
		tmp = (y / a) * t;
	} else {
		tmp = (t / a) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y * (z - t)) <= -5e-89:
		tmp = (y / a) * t
	else:
		tmp = (t / a) * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(y * Float64(z - t)) <= -5e-89)
		tmp = Float64(Float64(y / a) * t);
	else
		tmp = Float64(Float64(t / a) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y * (z - t)) <= -5e-89)
		tmp = (y / a) * t;
	else
		tmp = (t / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision], -5e-89], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -4.9999999999999997e-89

   1. Initial program 93.3%

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

   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 31.3%

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

   4. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\frac{t \cdot y}{a}} \]
   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 34.3%

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

   6. Applied rewrites 34.3%

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

   if -4.9999999999999997e-89 < (*.f64 y (-.f64 z t))

   1. Initial program 93.3%

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

   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 31.3%

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

   4. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\frac{t \cdot y}{a}} \]
   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 31.3%

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

   6. Applied rewrites 31.3%

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

Alternative 11: 31.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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 / a) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

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 31.3%

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

4. Applied rewrites 31.3%

   \[\leadsto \color{blue}{\frac{t \cdot y}{a}} \]
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 31.3%

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

6. Applied rewrites 31.3%

   \[\leadsto \frac{t}{a} \cdot \color{blue}{y} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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)))