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

Percentage Accurate: 93.2% → 99.0%

Time: 3.3s

Alternatives: 11

Speedup: 1.0×

• 74.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{y}}, t - z, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+62}:\\ \;\;\;\;x - \frac{1}{\frac{a}{\left(z - t\right) \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* y (- z t)) a))))
   (if (<= t_1 (- INFINITY))
     (fma (/ 1.0 (/ a y)) (- t z) x)
     (if (<= t_1 2e+62)
       (- x (/ 1.0 (/ a (* (- z t) y))))
       (fma (/ y a) (- t z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y * (z - t)) / a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((1.0 / (a / y)), (t - z), x);
	} else if (t_1 <= 2e+62) {
		tmp = x - (1.0 / (a / ((z - t) * y)));
	} else {
		tmp = fma((y / a), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y * Float64(z - t)) / a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(1.0 / Float64(a / y)), Float64(t - z), x);
	elseif (t_1 <= 2e+62)
		tmp = Float64(x - Float64(1.0 / Float64(a / Float64(Float64(z - t) * y))));
	else
		tmp = fma(Float64(y / a), Float64(t - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(1.0 / N[(a / y), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+62], N[(x - N[(1.0 / N[(a / N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 97.3%

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

   5. Applied rewrites 97.3%

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

   if -inf.0 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 2.0000000000000001e62

   1. Initial program 93.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 93.1%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 93.1%

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

   3. Applied rewrites 93.1%

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

   if 2.0000000000000001e62 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{y}}, t - z, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-153}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* y (- z t)) a))))
   (if (<= t_1 (- INFINITY))
     (fma (/ 1.0 (/ a y)) (- t z) x)
     (if (<= t_1 5e-153) t_1 (fma (/ y a) (- t z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y * (z - t)) / a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((1.0 / (a / y)), (t - z), x);
	} else if (t_1 <= 5e-153) {
		tmp = t_1;
	} else {
		tmp = fma((y / a), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y * Float64(z - t)) / a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(1.0 / Float64(a / y)), Float64(t - z), x);
	elseif (t_1 <= 5e-153)
		tmp = t_1;
	else
		tmp = fma(Float64(y / a), Float64(t - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(1.0 / N[(a / y), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e-153], t$95$1, N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 97.3%

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

   5. Applied rewrites 97.3%

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

   if -inf.0 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 5.0000000000000003e-153

   1. Initial program 93.2%

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

   if 5.0000000000000003e-153 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

Alternative 3: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-153}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* y (- z t)) a))))
   (if (<= t_1 -5e+303)
     (fma (/ (- t z) a) y x)
     (if (<= t_1 5e-153) t_1 (fma (/ y a) (- t z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y * (z - t)) / a);
	double tmp;
	if (t_1 <= -5e+303) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 5e-153) {
		tmp = t_1;
	} else {
		tmp = fma((y / a), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y * Float64(z - t)) / a))
	tmp = 0.0
	if (t_1 <= -5e+303)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 5e-153)
		tmp = t_1;
	else
		tmp = fma(Float64(y / a), Float64(t - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+303], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e-153], t$95$1, N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < -4.9999999999999997e303

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 93.1%

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

   3. Applied rewrites 93.1%

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

   if -4.9999999999999997e303 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 5.0000000000000003e-153

   1. Initial program 93.2%

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

   if 5.0000000000000003e-153 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. mult-flip-rev N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 97.5%

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

3. Applied rewrites 97.5%

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

Alternative 5: 93.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq 1.56 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.56e+139) (fma (/ (- t z) a) y x) (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.56e+139) {
		tmp = fma(((t - z) / a), y, x);
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.56e+139)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.56e+139], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5599999999999999e139

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 93.1%

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

   3. Applied rewrites 93.1%

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

   if 1.5599999999999999e139 < t

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 71.2%

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

Alternative 6: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ y a) (- t z))))
   (if (<= t_1 -5e+84) t_2 (if (<= t_1 2e+126) (fma (/ t a) y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (y / a) * (t - z);
	double tmp;
	if (t_1 <= -5e+84) {
		tmp = t_2;
	} else if (t_1 <= 2e+126) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(Float64(y / a) * Float64(t - z))
	tmp = 0.0
	if (t_1 <= -5e+84)
		tmp = t_2;
	elseif (t_1 <= 2e+126)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+84], t$95$2, If[LessEqual[t$95$1, 2e+126], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.0000000000000001e84 or 1.9999999999999998e126 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.5%

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 61.0%

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

   8. Applied rewrites 61.0%

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

   if -5.0000000000000001e84 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.9999999999999998e126

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 71.2%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 68.0%

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

   8. Applied rewrites 68.0%

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

Alternative 7: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(t - z\right)}{a}\\ t_2 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{t - z}{a} \cdot y\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- t z)) a)) (t_2 (/ (* y (- z t)) a)))
   (if (<= t_2 (- INFINITY))
     (* (/ (- t z) a) y)
     (if (<= t_2 -5e+84) t_1 (if (<= t_2 2e+126) (fma (/ t a) y x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (t - z)) / a;
	double t_2 = (y * (z - t)) / a;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((t - z) / a) * y;
	} else if (t_2 <= -5e+84) {
		tmp = t_1;
	} else if (t_2 <= 2e+126) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(t - z)) / a)
	t_2 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t - z) / a) * y);
	elseif (t_2 <= -5e+84)
		tmp = t_1;
	elseif (t_2 <= 2e+126)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, -5e+84], t$95$1, If[LessEqual[t$95$2, 2e+126], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.5%

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 57.6%

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

   8. Applied rewrites 57.6%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a) < -5.0000000000000001e84 or 1.9999999999999998e126 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.5%

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

   6. Applied rewrites 57.5%

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

   if -5.0000000000000001e84 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.9999999999999998e126

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 71.2%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 68.0%

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

   8. Applied rewrites 68.0%

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

Alternative 8: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{t - z}{a} \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+275}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ (- t z) a) y)))
   (if (<= t_1 -5e+275) t_2 (if (<= t_1 2e+236) (fma (/ y a) t x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = ((t - z) / a) * y;
	double tmp;
	if (t_1 <= -5e+275) {
		tmp = t_2;
	} else if (t_1 <= 2e+236) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(Float64(Float64(t - z) / a) * y)
	tmp = 0.0
	if (t_1 <= -5e+275)
		tmp = t_2;
	elseif (t_1 <= 2e+236)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+275], t$95$2, If[LessEqual[t$95$1, 2e+236], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.0000000000000003e275 or 2.0000000000000001e236 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.5%

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 57.6%

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

   8. Applied rewrites 57.6%

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

   if -5.0000000000000003e275 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2.0000000000000001e236

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 97.5%

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 71.2%

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

Alternative 9: 71.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. mult-flip-rev N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 97.5%

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

3. Applied rewrites 97.5%

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

4. Taylor expanded in z around 0

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

6. Applied rewrites 71.2%

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

Alternative 10: 35.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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 = (y / a) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 32.4%

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

4. Applied rewrites 32.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 35.2%

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

6. Applied rewrites 35.2%

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

Alternative 11: 32.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 32.4%

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

4. Applied rewrites 32.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 32.5%

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

6. Applied rewrites 32.5%

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

Reproduce

herbie shell --seed 2025204 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64
  (- x (/ (* y (- z t)) a)))