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

Percentage Accurate: 93.5% → 98.8%

Time: 3.6s

Alternatives: 12

Speedup: 1.1×

• 24.9% 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: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \frac{z - t}{a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+189}:\\ \;\;\;\;x + \frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))) (t_2 (/ (- z t) a)))
   (if (<= t_1 (- INFINITY))
     (* t_2 y)
     (if (<= t_1 5e+189) (+ x (/ t_1 a)) (fma t_2 y x)))))

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;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 * y;
	} else if (t_1 <= 5e+189) {
		tmp = x + (t_1 / a);
	} else {
		tmp = fma(t_2, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	t_2 = Float64(Float64(z - t) / a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_2 * y);
	elseif (t_1 <= 5e+189)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = fma(t_2, y, x);
	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[(z - t), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * y), $MachinePrecision], If[LessEqual[t$95$1, 5e+189], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * y + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -inf.0

   1. Initial program 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. fp-cancel-sub-sign N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 61.7

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

   3. Applied rewrites 61.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 93.2

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

   8. Applied rewrites 93.2%

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

   if -inf.0 < (*.f64 y (-.f64 z t)) < 5.0000000000000004e189

   1. Initial program 99.3%

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

   if 5.0000000000000004e189 < (*.f64 y (-.f64 z t))

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 98.9

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

   3. Applied rewrites 98.9%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+226}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, y, \left(-y\right) \cdot t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 (- INFINITY))
     (* (/ (- z t) a) y)
     (if (<= t_1 2e+226)
       (+ x (/ (fma z y (* (- y) t)) a))
       (fma (- z t) (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((z - t) / a) * y;
	} else if (t_1 <= 2e+226) {
		tmp = x + (fma(z, y, (-y * t)) / a);
	} else {
		tmp = fma((z - t), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(z - t) / a) * y);
	elseif (t_1 <= 2e+226)
		tmp = Float64(x + Float64(fma(z, y, Float64(Float64(-y) * t)) / a));
	else
		tmp = fma(Float64(z - t), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+226], N[(x + N[(N[(z * y + N[((-y) * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -inf.0

   1. Initial program 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. fp-cancel-sub-sign N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 61.7

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

   3. Applied rewrites 61.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 93.2

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

   8. Applied rewrites 93.2%

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

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1.99999999999999992e226

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 99.4

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

   3. Applied rewrites 99.4%

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

   if 1.99999999999999992e226 < (*.f64 y (-.f64 z t))

   1. Initial program 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. fp-cancel-sub-sign N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 76.1

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

   3. Applied rewrites 76.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 3: 97.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a 2e-68) (fma (- z t) (/ y a) x) (fma (/ (- z t) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 2e-68) {
		tmp = fma((z - t), (y / a), x);
	} else {
		tmp = fma(((z - t) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 2e-68)
		tmp = fma(Float64(z - t), Float64(y / a), x);
	else
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, 2e-68], N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.00000000000000013e-68

   1. Initial program 95.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. fp-cancel-sub-sign N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 93.5

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

   3. Applied rewrites 93.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if 2.00000000000000013e-68 < a

   1. Initial program 89.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 98.7

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

   3. Applied rewrites 98.7%

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

Alternative 4: 97.3% 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 93.5%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. negate-sub N/A

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

   4. mul-1-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. fp-cancel-sub-sign N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 92.0

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

3. Applied rewrites 92.0%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. +-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower-/.f64 97.3

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

5. Applied rewrites 97.3%

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

Alternative 5: 86.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a} \cdot y\\ t_2 := \frac{\left(z - t\right) \cdot y}{a}\\ t_3 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{+69}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -5e+69)
       t_2
       (if (<= t_3 2e+34) (fma (/ z a) y x) (if (<= t_3 5e+306) 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 <= -5e+69) {
		tmp = t_2;
	} else if (t_3 <= 2e+34) {
		tmp = fma((z / a), y, x);
	} else if (t_3 <= 5e+306) {
		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 <= -5e+69)
		tmp = t_2;
	elseif (t_3 <= 2e+34)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_3 <= 5e+306)
		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, -5e+69], t$95$2, If[LessEqual[t$95$3, 2e+34], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$3, 5e+306], 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 4.99999999999999993e306 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. fp-cancel-sub-sign N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 75.9

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

   3. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 96.3

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

   8. Applied rewrites 96.3%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a) < -5.00000000000000036e69 or 1.99999999999999989e34 < (/.f64 (*.f64 y (-.f64 z t)) a) < 4.99999999999999993e306

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a}} \]
   3. 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 77.4

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

   4. Applied rewrites 77.4%

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

   if -5.00000000000000036e69 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.99999999999999989e34

   1. Initial program 99.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 98.6

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

   3. Applied rewrites 98.6%

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

   4. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 86.8

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

   6. Applied rewrites 86.8%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 85.3

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

   9. Applied rewrites 85.3%

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

Alternative 6: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+69}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (/ (* (- z t) y) a)))
   (if (<= t_1 -5e+69) t_2 (if (<= t_1 2e+34) (fma (/ z 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 = ((z - t) * y) / a;
	double tmp;
	if (t_1 <= -5e+69) {
		tmp = t_2;
	} else if (t_1 <= 2e+34) {
		tmp = fma((z / 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(z - t) * y) / a)
	tmp = 0.0
	if (t_1 <= -5e+69)
		tmp = t_2;
	elseif (t_1 <= 2e+34)
		tmp = fma(Float64(z / 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[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+69], t$95$2, If[LessEqual[t$95$1, 2e+34], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.00000000000000036e69 or 1.99999999999999989e34 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 89.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a}} \]
   3. 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 79.2

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

   4. Applied rewrites 79.2%

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

   if -5.00000000000000036e69 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.99999999999999989e34

   1. Initial program 99.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 98.6

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

   3. Applied rewrites 98.6%

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

   4. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 86.8

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

   6. Applied rewrites 86.8%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 85.3

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

   9. Applied rewrites 85.3%

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

Alternative 7: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{t \cdot y}{a}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* t y) a))))
   (if (<= t -7.6e+123) t_1 (if (<= t 1e+60) (fma (/ y a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((t * y) / a);
	double tmp;
	if (t <= -7.6e+123) {
		tmp = t_1;
	} else if (t <= 1e+60) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(t * y) / a))
	tmp = 0.0
	if (t <= -7.6e+123)
		tmp = t_1;
	elseif (t <= 1e+60)
		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[(x - N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.6e+123], t$95$1, If[LessEqual[t, 1e+60], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.59999999999999989e123 or 9.9999999999999995e59 < t

   1. Initial program 90.9%

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

   2. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. negate-sub N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 82.5

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

   4. Applied rewrites 82.5%

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

   if -7.59999999999999989e123 < t < 9.9999999999999995e59

   1. Initial program 94.9%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{a}} \]
   3. 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 84.2

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

   4. Applied rewrites 84.2%

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

Alternative 8: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+156}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.4e+156)
   (* (/ y a) (- t))
   (if (<= t 6e+209) (fma (/ y a) z x) (/ (* (- t) y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.4e+156) {
		tmp = (y / a) * -t;
	} else if (t <= 6e+209) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = (-t * y) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.4e+156)
		tmp = Float64(Float64(y / a) * Float64(-t));
	elseif (t <= 6e+209)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(Float64(Float64(-t) * y) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.4e+156], N[(N[(y / a), $MachinePrecision] * (-t)), $MachinePrecision], If[LessEqual[t, 6e+209], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[((-t) * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.4000000000000001e156

   1. Initial program 90.1%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{a}} \]
   3. 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 65.3

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

   4. Applied rewrites 65.3%

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

   if -3.4000000000000001e156 < t < 5.99999999999999971e209

   1. Initial program 94.5%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{a}} \]
   3. 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 79.1

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

   4. Applied rewrites 79.1%

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

   if 5.99999999999999971e209 < t

   1. Initial program 88.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 87.1

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

   3. Applied rewrites 87.1%

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

   4. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-*.f64 66.5

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

   6. Applied rewrites 66.5%

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

Alternative 9: 59.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-33}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 -5e-33) (* z (/ y a)) (if (<= t_1 2e+34) x (* (/ y a) (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -5e-33) {
		tmp = z * (y / a);
	} else if (t_1 <= 2e+34) {
		tmp = x;
	} else {
		tmp = (y / a) * -t;
	}
	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 = (y * (z - t)) / a
    if (t_1 <= (-5d-33)) then
        tmp = z * (y / a)
    else if (t_1 <= 2d+34) then
        tmp = x
    else
        tmp = (y / a) * -t
    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 tmp;
	if (t_1 <= -5e-33) {
		tmp = z * (y / a);
	} else if (t_1 <= 2e+34) {
		tmp = x;
	} else {
		tmp = (y / a) * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if t_1 <= -5e-33:
		tmp = z * (y / a)
	elif t_1 <= 2e+34:
		tmp = x
	else:
		tmp = (y / a) * -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= -5e-33)
		tmp = Float64(z * Float64(y / a));
	elseif (t_1 <= 2e+34)
		tmp = x;
	else
		tmp = Float64(Float64(y / a) * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	tmp = 0.0;
	if (t_1 <= -5e-33)
		tmp = z * (y / a);
	elseif (t_1 <= 2e+34)
		tmp = x;
	else
		tmp = (y / a) * -t;
	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]}, If[LessEqual[t$95$1, -5e-33], N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+34], x, N[(N[(y / a), $MachinePrecision] * (-t)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.00000000000000028e-33

   1. Initial program 90.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 89.2

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

   3. Applied rewrites 89.2%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-/.f64 72.8

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

   6. Applied rewrites 73.9%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 40.8%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 46.3

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

   11. Applied rewrites 46.3%

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

   if -5.00000000000000028e-33 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.99999999999999989e34

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 77.8%

      \[\leadsto \color{blue}{x} \]

   if 1.99999999999999989e34 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 90.2%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot y}{a}} \]
   3. 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 50.5

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

   4. Applied rewrites 50.5%

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

Alternative 10: 57.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := z \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* z (/ y a))))
   (if (<= t_1 -5e-33) t_2 (if (<= t_1 5e-35) 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 = z * (y / a);
	double tmp;
	if (t_1 <= -5e-33) {
		tmp = t_2;
	} else if (t_1 <= 5e-35) {
		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 = z * (y / a)
    if (t_1 <= (-5d-33)) then
        tmp = t_2
    else if (t_1 <= 5d-35) 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 = z * (y / a);
	double tmp;
	if (t_1 <= -5e-33) {
		tmp = t_2;
	} else if (t_1 <= 5e-35) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = z * (y / a)
	tmp = 0
	if t_1 <= -5e-33:
		tmp = t_2
	elif t_1 <= 5e-35:
		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(z * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -5e-33)
		tmp = t_2;
	elseif (t_1 <= 5e-35)
		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 = z * (y / a);
	tmp = 0.0;
	if (t_1 <= -5e-33)
		tmp = t_2;
	elseif (t_1 <= 5e-35)
		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[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-33], t$95$2, If[LessEqual[t$95$1, 5e-35], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.00000000000000028e-33 or 4.99999999999999964e-35 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 89.7

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

   3. Applied rewrites 89.7%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-/.f64 74.0

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

   6. Applied rewrites 75.2%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 41.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 46.1

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

   11. Applied rewrites 46.1%

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

   if -5.00000000000000028e-33 < (/.f64 (*.f64 y (-.f64 z t)) a) < 4.99999999999999964e-35

   1. Initial program 98.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 81.6%

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

Alternative 11: 56.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := y \cdot \frac{z}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+53}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* y (/ z a))))
   (if (<= t_1 -4e+53) t_2 (if (<= t_1 5e-35) 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 * (z / a);
	double tmp;
	if (t_1 <= -4e+53) {
		tmp = t_2;
	} else if (t_1 <= 5e-35) {
		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 * (z / a)
    if (t_1 <= (-4d+53)) then
        tmp = t_2
    else if (t_1 <= 5d-35) 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 * (z / a);
	double tmp;
	if (t_1 <= -4e+53) {
		tmp = t_2;
	} else if (t_1 <= 5e-35) {
		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 * (z / a)
	tmp = 0
	if t_1 <= -4e+53:
		tmp = t_2
	elif t_1 <= 5e-35:
		tmp = x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(y * Float64(z / a))
	tmp = 0.0
	if (t_1 <= -4e+53)
		tmp = t_2;
	elseif (t_1 <= 5e-35)
		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 * (z / a);
	tmp = 0.0;
	if (t_1 <= -4e+53)
		tmp = t_2;
	elseif (t_1 <= 5e-35)
		tmp = x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+53], t$95$2, If[LessEqual[t$95$1, 5e-35], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4e53 or 4.99999999999999964e-35 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 90.1%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 43.0

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

   4. Applied rewrites 43.0%

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

   if -4e53 < (/.f64 (*.f64 y (-.f64 z t)) a) < 4.99999999999999964e-35

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 77.6%

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

Alternative 12: 39.4% accurate, 12.7× 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 93.5%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 39.4%

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

Reproduce

herbie shell --seed 2025110 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64
  (+ x (/ (* y (- z t)) a)))