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

Percentage Accurate: 93.4% → 97.4%

Time: 2.8s

Alternatives: 11

Speedup: 1.1×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 93.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y * (z - t)) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lower-/.f64 97.4

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

4. Applied rewrites 97.4%

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

Alternative 2: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a} \cdot y\\ t_2 := \frac{\left(z - t\right) \cdot y}{a}\\ t_3 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+142}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+288}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) a) y))
        (t_2 (/ (* (- z t) y) a))
        (t_3 (/ (* y (- z t)) a)))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -2e+142)
       t_2
       (if (<= t_3 5e+55)
         (fma (- t) (/ y a) x)
         (if (<= t_3 5e+288) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / a) * y;
	double t_2 = ((z - t) * y) / a;
	double t_3 = (y * (z - t)) / a;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -2e+142) {
		tmp = t_2;
	} else if (t_3 <= 5e+55) {
		tmp = fma(-t, (y / a), x);
	} else if (t_3 <= 5e+288) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) / a) * y)
	t_2 = Float64(Float64(Float64(z - t) * y) / a)
	t_3 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -2e+142)
		tmp = t_2;
	elseif (t_3 <= 5e+55)
		tmp = fma(Float64(-t), Float64(y / a), x);
	elseif (t_3 <= 5e+288)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -2e+142], t$95$2, If[LessEqual[t$95$3, 5e+55], N[((-t) * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$3, 5e+288], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -inf.0 or 5.0000000000000003e288 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 81.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 95.4

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

   7. Applied rewrites 95.4%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a) < -2.0000000000000001e142 or 5.00000000000000046e55 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.0000000000000003e288

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -2.0000000000000001e142 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.00000000000000046e55

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.9

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

   7. Applied rewrites 82.9%

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

Alternative 3: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a} \cdot y\\ t_2 := \frac{\left(z - t\right) \cdot y}{a}\\ t_3 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+288}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) a) y))
        (t_2 (/ (* (- z t) y) a))
        (t_3 (/ (* y (- z t)) a)))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -1e+144)
       t_2
       (if (<= t_3 5e+55) (- x (/ (* y t) a)) (if (<= t_3 5e+288) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / a) * y;
	double t_2 = ((z - t) * y) / a;
	double t_3 = (y * (z - t)) / a;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -1e+144) {
		tmp = t_2;
	} else if (t_3 <= 5e+55) {
		tmp = x - ((y * t) / a);
	} else if (t_3 <= 5e+288) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / a) * y;
	double t_2 = ((z - t) * y) / a;
	double t_3 = (y * (z - t)) / a;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_3 <= -1e+144) {
		tmp = t_2;
	} else if (t_3 <= 5e+55) {
		tmp = x - ((y * t) / a);
	} else if (t_3 <= 5e+288) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((z - t) / a) * y
	t_2 = ((z - t) * y) / a
	t_3 = (y * (z - t)) / a
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_1
	elif t_3 <= -1e+144:
		tmp = t_2
	elif t_3 <= 5e+55:
		tmp = x - ((y * t) / a)
	elif t_3 <= 5e+288:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) / a) * y)
	t_2 = Float64(Float64(Float64(z - t) * y) / a)
	t_3 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -1e+144)
		tmp = t_2;
	elseif (t_3 <= 5e+55)
		tmp = Float64(x - Float64(Float64(y * t) / a));
	elseif (t_3 <= 5e+288)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z - t) / a) * y;
	t_2 = ((z - t) * y) / a;
	t_3 = (y * (z - t)) / a;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_1;
	elseif (t_3 <= -1e+144)
		tmp = t_2;
	elseif (t_3 <= 5e+55)
		tmp = x - ((y * t) / a);
	elseif (t_3 <= 5e+288)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -1e+144], t$95$2, If[LessEqual[t$95$3, 5e+55], N[(x - N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+288], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -inf.0 or 5.0000000000000003e288 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 81.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 95.4

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

   7. Applied rewrites 95.4%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a) < -1.00000000000000002e144 or 5.00000000000000046e55 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.0000000000000003e288

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if -1.00000000000000002e144 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.00000000000000046e55

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 82.7

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

   5. Applied rewrites 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 84.0

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

   7. Applied rewrites 84.0%

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

Alternative 4: 82.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{z - t}{a} \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+233}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ (- z t) a) y)))
   (if (<= t_1 -5e+233) t_2 (if (<= t_1 5e+55) (- x (/ (* y t) a)) 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 = ((z - t) / a) * y;
	double tmp;
	if (t_1 <= -5e+233) {
		tmp = t_2;
	} else if (t_1 <= 5e+55) {
		tmp = x - ((y * t) / a);
	} 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 = ((z - t) / a) * y
    if (t_1 <= (-5d+233)) then
        tmp = t_2
    else if (t_1 <= 5d+55) then
        tmp = x - ((y * t) / a)
    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 = ((z - t) / a) * y;
	double tmp;
	if (t_1 <= -5e+233) {
		tmp = t_2;
	} else if (t_1 <= 5e+55) {
		tmp = x - ((y * t) / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = ((z - t) / a) * y
	tmp = 0
	if t_1 <= -5e+233:
		tmp = t_2
	elif t_1 <= 5e+55:
		tmp = x - ((y * t) / a)
	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(z - t) / a) * y)
	tmp = 0.0
	if (t_1 <= -5e+233)
		tmp = t_2;
	elseif (t_1 <= 5e+55)
		tmp = Float64(x - Float64(Float64(y * t) / a));
	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 = ((z - t) / a) * y;
	tmp = 0.0;
	if (t_1 <= -5e+233)
		tmp = t_2;
	elseif (t_1 <= 5e+55)
		tmp = x - ((y * t) / a);
	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[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+233], t$95$2, If[LessEqual[t$95$1, 5e+55], N[(x - N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.00000000000000009e233 or 5.00000000000000046e55 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 86.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 83.8

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

   7. Applied rewrites 83.8%

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

   if -5.00000000000000009e233 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.00000000000000046e55

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.8

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

   7. Applied rewrites 81.8%

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

Alternative 5: 82.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{z - t}{a} \cdot y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+126}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ (- z t) a) y)))
   (if (<= t_1 -2e+126) t_2 (if (<= t_1 5e+55) (- x (* y (/ t a))) 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 = ((z - t) / a) * y;
	double tmp;
	if (t_1 <= -2e+126) {
		tmp = t_2;
	} else if (t_1 <= 5e+55) {
		tmp = x - (y * (t / a));
	} 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 = ((z - t) / a) * y
    if (t_1 <= (-2d+126)) then
        tmp = t_2
    else if (t_1 <= 5d+55) then
        tmp = x - (y * (t / a))
    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 = ((z - t) / a) * y;
	double tmp;
	if (t_1 <= -2e+126) {
		tmp = t_2;
	} else if (t_1 <= 5e+55) {
		tmp = x - (y * (t / a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = ((z - t) / a) * y
	tmp = 0
	if t_1 <= -2e+126:
		tmp = t_2
	elif t_1 <= 5e+55:
		tmp = x - (y * (t / a))
	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(z - t) / a) * y)
	tmp = 0.0
	if (t_1 <= -2e+126)
		tmp = t_2;
	elseif (t_1 <= 5e+55)
		tmp = Float64(x - Float64(y * Float64(t / a)));
	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 = ((z - t) / a) * y;
	tmp = 0.0;
	if (t_1 <= -2e+126)
		tmp = t_2;
	elseif (t_1 <= 5e+55)
		tmp = x - (y * (t / a));
	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[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+126], t$95$2, If[LessEqual[t$95$1, 5e+55], N[(x - N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1.99999999999999985e126 or 5.00000000000000046e55 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 80.9

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

   7. Applied rewrites 80.9%

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

   if -1.99999999999999985e126 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.00000000000000046e55

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.4

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

   5. Applied rewrites 83.4%

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

Alternative 6: 59.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{y}{a} \cdot z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+134}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ y a) z)))
   (if (<= t_1 -5e+134) t_2 (if (<= t_1 5e+55) 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) * z;
	double tmp;
	if (t_1 <= -5e+134) {
		tmp = t_2;
	} else if (t_1 <= 5e+55) {
		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) * z
    if (t_1 <= (-5d+134)) then
        tmp = t_2
    else if (t_1 <= 5d+55) 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) * z;
	double tmp;
	if (t_1 <= -5e+134) {
		tmp = t_2;
	} else if (t_1 <= 5e+55) {
		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) * z
	tmp = 0
	if t_1 <= -5e+134:
		tmp = t_2
	elif t_1 <= 5e+55:
		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) * z)
	tmp = 0.0
	if (t_1 <= -5e+134)
		tmp = t_2;
	elseif (t_1 <= 5e+55)
		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) * z;
	tmp = 0.0;
	if (t_1 <= -5e+134)
		tmp = t_2;
	elseif (t_1 <= 5e+55)
		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] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+134], t$95$2, If[LessEqual[t$95$1, 5e+55], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4.99999999999999981e134 or 5.00000000000000046e55 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 50.9

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

   7. Applied rewrites 50.9%

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

   if -4.99999999999999981e134 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.00000000000000046e55

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 69.0%

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

Alternative 7: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \frac{z - t}{a} \cdot y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+65}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))) (t_2 (* (/ (- z t) a) y)))
   (if (<= t_1 -1e+65) t_2 (if (<= t_1 2e+77) (fma (/ y a) z x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double t_2 = ((z - t) / a) * y;
	double tmp;
	if (t_1 <= -1e+65) {
		tmp = t_2;
	} else if (t_1 <= 2e+77) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	t_2 = Float64(Float64(Float64(z - t) / a) * y)
	tmp = 0.0
	if (t_1 <= -1e+65)
		tmp = t_2;
	elseif (t_1 <= 2e+77)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+65], t$95$2, If[LessEqual[t$95$1, 2e+77], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -9.9999999999999999e64 or 1.99999999999999997e77 < (*.f64 y (-.f64 z t))

   1. Initial program 88.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 79.2

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

   7. Applied rewrites 79.2%

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

   if -9.9999999999999999e64 < (*.f64 y (-.f64 z t)) < 1.99999999999999997e77

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

         \[\leadsto \frac{y}{a} \cdot z + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 79.7

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

   5. Applied rewrites 79.7%

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

Alternative 8: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-t\right)\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y a) (- t))))
   (if (<= t -2.05e+86) t_1 (if (<= t 2.8e+91) (fma (/ y a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / a) * -t;
	double tmp;
	if (t <= -2.05e+86) {
		tmp = t_1;
	} else if (t <= 2.8e+91) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / a) * Float64(-t))
	tmp = 0.0
	if (t <= -2.05e+86)
		tmp = t_1;
	elseif (t <= 2.8e+91)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * (-t)), $MachinePrecision]}, If[LessEqual[t, -2.05e+86], t$95$1, If[LessEqual[t, 2.8e+91], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.05e86 or 2.7999999999999999e91 < t

   1. Initial program 90.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if -2.05e86 < t < 2.7999999999999999e91

   1. Initial program 94.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

         \[\leadsto \frac{y}{a} \cdot z + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

Alternative 9: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-y\right) \cdot \frac{t}{a}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+214}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- y) (/ t a))))
   (if (<= t -8.5e+214) t_1 (if (<= t 3e+92) (fma (/ y a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -y * (t / a);
	double tmp;
	if (t <= -8.5e+214) {
		tmp = t_1;
	} else if (t <= 3e+92) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-y) * Float64(t / a))
	tmp = 0.0
	if (t <= -8.5e+214)
		tmp = t_1;
	elseif (t <= 3e+92)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-y) * N[(t / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e+214], t$95$1, If[LessEqual[t, 3e+92], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.50000000000000045e214 or 3.00000000000000013e92 < t

   1. Initial program 90.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-/.f64 59.6

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

   7. Applied rewrites 59.6%

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

   if -8.50000000000000045e214 < t < 3.00000000000000013e92

   1. Initial program 94.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

         \[\leadsto \frac{y}{a} \cdot z + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.5

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

   5. Applied rewrites 80.5%

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

Alternative 10: 71.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

      \[\leadsto \frac{y}{a} \cdot z + x \]

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 71.3

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

5. Applied rewrites 71.3%

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

Alternative 11: 39.6% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

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 93.4%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 39.6%

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

Developer Target 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{t\_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (+ x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (+ x (/ (* y (- z t)) a))
       (+ x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x + (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) / a)
    else
        tmp = x + (y / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x + (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) / a)
	else:
		tmp = x + (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x + Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x + Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x + (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) / a);
	else
		tmp = x + (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x + N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2025091 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -430450648655599/4000000000000000000000000) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t)))))))

  (+ x (/ (* y (- z t)) a)))