Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, H

Percentage Accurate: 95.5% → 97.3%

Time: 7.1s

Alternatives: 14

Speedup: 1.0×

• 28.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{-265}:\\ \;\;\;\;x - \frac{t\_1}{3 \cdot z}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{t\_1}{z}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (/ t y))))
   (if (<= y -9.2e-265)
     (- x (/ t_1 (* 3.0 z)))
     (if (<= y 2.6e-80)
       (fma (/ 0.3333333333333333 y) (/ t z) x)
       (- x (/ (/ t_1 z) 3.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y - (t / y);
	double tmp;
	if (y <= -9.2e-265) {
		tmp = x - (t_1 / (3.0 * z));
	} else if (y <= 2.6e-80) {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	} else {
		tmp = x - ((t_1 / z) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y - Float64(t / y))
	tmp = 0.0
	if (y <= -9.2e-265)
		tmp = Float64(x - Float64(t_1 / Float64(3.0 * z)));
	elseif (y <= 2.6e-80)
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	else
		tmp = Float64(x - Float64(Float64(t_1 / z) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.2e-265], N[(x - N[(t$95$1 / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e-80], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(t$95$1 / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.1999999999999996e-265

   1. Initial program 94.3%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]

   if -9.1999999999999996e-265 < y < 2.6000000000000001e-80

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

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

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

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

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} + x \]

      16. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{y} \cdot \frac{t}{z}} + x \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{3}}{y}, \frac{t}{z}, x\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{y}}, \frac{t}{z}, x\right) \]

      19. lower-/.f64 94.7

          \[\leadsto \mathsf{fma}\left(\frac{0.3333333333333333}{y}, \color{blue}{\frac{t}{z}}, x\right) \]

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)} \]

   if 2.6000000000000001e-80 < y

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      3. *-commutative N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

      6. lower-/.f64 99.8

         \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{z}}}{3} \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{\frac{y - \frac{t}{y}}{z}}{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-265}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y)))))
   (if (<= t_1 2e+295) t_1 (/ (* 0.3333333333333333 (- (/ t y) y)) z))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	double tmp;
	if (t_1 <= 2e+295) {
		tmp = t_1;
	} else {
		tmp = (0.3333333333333333 * ((t / y) - y)) / z;
	}
	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)
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) :: t_1
    real(8) :: tmp
    t_1 = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
    if (t_1 <= 2d+295) then
        tmp = t_1
    else
        tmp = (0.3333333333333333d0 * ((t / y) - y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	double tmp;
	if (t_1 <= 2e+295) {
		tmp = t_1;
	} else {
		tmp = (0.3333333333333333 * ((t / y) - y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))
	tmp = 0
	if t_1 <= 2e+295:
		tmp = t_1
	else:
		tmp = (0.3333333333333333 * ((t / y) - y)) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
	tmp = 0.0
	if (t_1 <= 2e+295)
		tmp = t_1;
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(Float64(t / y) - y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	tmp = 0.0;
	if (t_1 <= 2e+295)
		tmp = t_1;
	else
		tmp = (0.3333333333333333 * ((t / y) - y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+295], t$95$1, N[(N[(0.3333333333333333 * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2e295

   1. Initial program 97.5%

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

   if 2e295 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\frac{t}{y \cdot z} - \frac{y}{z}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{t}{y \cdot z} - \frac{y}{z}\right) \cdot \frac{1}{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{t}{y \cdot z} - \frac{y}{z}\right) \cdot \frac{1}{3}} \]

      4. associate-/r* N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{t}{y}}{z}} - \frac{y}{z}\right) \cdot \frac{1}{3} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{y} - y}{z}} \cdot \frac{1}{3} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{y} - y}{z}} \cdot \frac{1}{3} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{t}{y} - y}}{z} \cdot \frac{1}{3} \]

      8. lower-/.f64 99.9

         \[\leadsto \frac{\color{blue}{\frac{t}{y}} - y}{z} \cdot 0.3333333333333333 \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{t}{y} - y}{z} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{\color{blue}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t\_1 + \frac{t}{\left(z \cdot 3\right) \cdot y} \leq 2 \cdot 10^{+295}:\\ \;\;\;\;t\_1 + \frac{t}{\left(3 \cdot y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= (+ t_1 (/ t (* (* z 3.0) y))) 2e+295)
     (+ t_1 (/ t (* (* 3.0 y) z)))
     (/ (* 0.3333333333333333 (- (/ t y) y)) z))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((t_1 + (t / ((z * 3.0) * y))) <= 2e+295) {
		tmp = t_1 + (t / ((3.0 * y) * z));
	} else {
		tmp = (0.3333333333333333 * ((t / y) - y)) / z;
	}
	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)
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) :: t_1
    real(8) :: tmp
    t_1 = x - (y / (z * 3.0d0))
    if ((t_1 + (t / ((z * 3.0d0) * y))) <= 2d+295) then
        tmp = t_1 + (t / ((3.0d0 * y) * z))
    else
        tmp = (0.3333333333333333d0 * ((t / y) - y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((t_1 + (t / ((z * 3.0) * y))) <= 2e+295) {
		tmp = t_1 + (t / ((3.0 * y) * z));
	} else {
		tmp = (0.3333333333333333 * ((t / y) - y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / (z * 3.0))
	tmp = 0
	if (t_1 + (t / ((z * 3.0) * y))) <= 2e+295:
		tmp = t_1 + (t / ((3.0 * y) * z))
	else:
		tmp = (0.3333333333333333 * ((t / y) - y)) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (Float64(t_1 + Float64(t / Float64(Float64(z * 3.0) * y))) <= 2e+295)
		tmp = Float64(t_1 + Float64(t / Float64(Float64(3.0 * y) * z)));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(Float64(t / y) - y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / (z * 3.0));
	tmp = 0.0;
	if ((t_1 + (t / ((z * 3.0) * y))) <= 2e+295)
		tmp = t_1 + (t / ((3.0 * y) * z));
	else
		tmp = (0.3333333333333333 * ((t / y) - y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+295], N[(t$95$1 + N[(t / N[(N[(3.0 * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2e295

   1. Initial program 97.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right)} \cdot y} \]

      3. associate-*l* N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(3 \cdot y\right) \cdot z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(3 \cdot y\right) \cdot z}} \]

      6. lower-*.f64 97.5

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(3 \cdot y\right)} \cdot z} \]

   4. Applied rewrites 97.5%

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(3 \cdot y\right) \cdot z}} \]

   if 2e295 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\frac{t}{y \cdot z} - \frac{y}{z}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{t}{y \cdot z} - \frac{y}{z}\right) \cdot \frac{1}{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{t}{y \cdot z} - \frac{y}{z}\right) \cdot \frac{1}{3}} \]

      4. associate-/r* N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{t}{y}}{z}} - \frac{y}{z}\right) \cdot \frac{1}{3} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{y} - y}{z}} \cdot \frac{1}{3} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{y} - y}{z}} \cdot \frac{1}{3} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{t}{y} - y}}{z} \cdot \frac{1}{3} \]

      8. lower-/.f64 99.9

         \[\leadsto \frac{\color{blue}{\frac{t}{y}} - y}{z} \cdot 0.3333333333333333 \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{t}{y} - y}{z} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{\color{blue}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(3 \cdot y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{3 \cdot z}}{y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ (/ t (* 3.0 z)) y)))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (3.0 * z)) / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + ((t / (3.0d0 * z)) / y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (3.0 * z)) / y);
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + ((t / (3.0 * z)) / y)

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / Float64(3.0 * z)) / y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + ((t / (3.0 * z)) / y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(3.0 * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

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

   1. lift-/.f64 N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}} \]

   3. associate-/r* N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

   5. lower-/.f64 97.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{t}{z \cdot 3}}}{y} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{\color{blue}{z \cdot 3}}}{y} \]

   7. *-commutative N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{\color{blue}{3 \cdot z}}}{y} \]

   8. lower-*.f64 97.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{\color{blue}{3 \cdot z}}}{y} \]

4. Applied rewrites 97.5%

   \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{3 \cdot z}}{y}} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-265} \lor \neg \left(y \leq 2.6 \cdot 10^{-80}\right):\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -9.2e-265) (not (<= y 2.6e-80)))
   (- x (/ (- y (/ t y)) (* 3.0 z)))
   (fma (/ 0.3333333333333333 y) (/ t z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -9.2e-265) || !(y <= 2.6e-80)) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -9.2e-265) || !(y <= 2.6e-80))
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -9.2e-265], N[Not[LessEqual[y, 2.6e-80]], $MachinePrecision]], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.1999999999999996e-265 or 2.6000000000000001e-80 < y

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]

   if -9.1999999999999996e-265 < y < 2.6000000000000001e-80

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

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

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

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

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} + x \]

      16. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{y} \cdot \frac{t}{z}} + x \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{3}}{y}, \frac{t}{z}, x\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{y}}, \frac{t}{z}, x\right) \]

      19. lower-/.f64 94.7

          \[\leadsto \mathsf{fma}\left(\frac{0.3333333333333333}{y}, \color{blue}{\frac{t}{z}}, x\right) \]

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-265} \lor \neg \left(y \leq 2.6 \cdot 10^{-80}\right):\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-265} \lor \neg \left(y \leq 2.6 \cdot 10^{-80}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -9.2e-265) (not (<= y 2.6e-80)))
   (fma (/ (- (/ t y) y) z) 0.3333333333333333 x)
   (fma (/ 0.3333333333333333 y) (/ t z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -9.2e-265) || !(y <= 2.6e-80)) {
		tmp = fma((((t / y) - y) / z), 0.3333333333333333, x);
	} else {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -9.2e-265) || !(y <= 2.6e-80))
		tmp = fma(Float64(Float64(Float64(t / y) - y) / z), 0.3333333333333333, x);
	else
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -9.2e-265], N[Not[LessEqual[y, 2.6e-80]], $MachinePrecision]], N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.1999999999999996e-265 or 2.6000000000000001e-80 < y

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)} \]

   if -9.1999999999999996e-265 < y < 2.6000000000000001e-80

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

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

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

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

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} + x \]

      16. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{y} \cdot \frac{t}{z}} + x \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{3}}{y}, \frac{t}{z}, x\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{y}}, \frac{t}{z}, x\right) \]

      19. lower-/.f64 94.7

          \[\leadsto \mathsf{fma}\left(\frac{0.3333333333333333}{y}, \color{blue}{\frac{t}{z}}, x\right) \]

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-265} \lor \neg \left(y \leq 2.6 \cdot 10^{-80}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.8e-86)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 1.5e-21)
     (fma (/ 0.3333333333333333 y) (/ t z) x)
     (- x (/ (/ y z) 3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.8e-86) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 1.5e-21) {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	} else {
		tmp = x - ((y / z) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.8e-86)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 1.5e-21)
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	else
		tmp = Float64(x - Float64(Float64(y / z) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.8e-86], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.5e-21], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.79999999999999983e-86

   1. Initial program 95.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \cdot y \]

      3. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \frac{x}{y}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \cdot y \]

      6. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \cdot y \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \cdot y \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot -1\right)} \cdot y \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(-1 \cdot y\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y\right)} \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right) \cdot y} \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      14. +-commutative N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)}\right)\right) \]

      15. distribute-neg-in N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right)} \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}} + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1}\right) \]

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

          \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} - \frac{x}{y} \cdot -1\right)} \]

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -1.79999999999999983e-86 < y < 1.49999999999999996e-21

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

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

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

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

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot t}{y \cdot z}} + x \]

      16. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{y} \cdot \frac{t}{z}} + x \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{3}}{y}, \frac{t}{z}, x\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{3}}{y}}, \frac{t}{z}, x\right) \]

      19. lower-/.f64 95.9

          \[\leadsto \mathsf{fma}\left(\frac{0.3333333333333333}{y}, \color{blue}{\frac{t}{z}}, x\right) \]

   5. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)} \]

   if 1.49999999999999996e-21 < y

   1. Initial program 97.3%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      3. *-commutative N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

      6. lower-/.f64 99.8

         \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{z}}}{3} \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 89.1

         \[\leadsto x - \frac{\color{blue}{\frac{y}{z}}}{3} \]

   9. Applied rewrites 89.1%

      \[\leadsto x - \frac{\color{blue}{\frac{y}{z}}}{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.8e-86)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 5.6e+65)
     (fma (/ t (* z y)) 0.3333333333333333 x)
     (- x (/ (/ y z) 3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.8e-86) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 5.6e+65) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = x - ((y / z) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.8e-86)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 5.6e+65)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = Float64(x - Float64(Float64(y / z) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.8e-86], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 5.6e+65], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(x - N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.79999999999999983e-86

   1. Initial program 95.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \cdot y \]

      3. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \frac{x}{y}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \cdot y \]

      6. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \cdot y \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \cdot y \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot -1\right)} \cdot y \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(-1 \cdot y\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y\right)} \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right) \cdot y} \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      14. +-commutative N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)}\right)\right) \]

      15. distribute-neg-in N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right)} \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}} + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1}\right) \]

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

          \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} - \frac{x}{y} \cdot -1\right)} \]

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -1.79999999999999983e-86 < y < 5.5999999999999998e65

   1. Initial program 94.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}} \]

      3. associate-/r* N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

      5. lower-/.f64 97.3

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{t}{z \cdot 3}}}{y} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{\color{blue}{z \cdot 3}}}{y} \]

      7. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{\color{blue}{3 \cdot z}}}{y} \]

      8. lower-*.f64 97.3

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{\color{blue}{3 \cdot z}}}{y} \]

   4. Applied rewrites 97.3%

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{3 \cdot z}}{y}} \]

   5. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \frac{t}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot y}}{y} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \frac{t}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}}{y} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y} \]

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

         \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot y\right)}}{y} \]

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

         \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)}\right)}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right)\right)}{y} \]

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

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot x}\right)}{y} \]

      12. distribute-neg-frac N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot x}{y}\right)\right)} \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot x}{\mathsf{neg}\left(y\right)}} \]

      14. associate-*r/ N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{x}{\mathsf{neg}\left(y\right)}} \]

      15. distribute-neg-frac2 N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)} \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{x}{y}\right)\right)} \]

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

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \cdot \frac{x}{y}} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{y} \cdot \frac{x}{y} \]

      19. associate-/l* N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\frac{y \cdot x}{y}} \]

      20. *-commutative N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\color{blue}{x \cdot y}}{y} \]

      21. associate-/l* N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{x \cdot \frac{y}{y}} \]

      22. *-inverses N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + x \cdot \color{blue}{1} \]

      23. *-rgt-identity N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{x} \]

   7. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)} \]

   if 5.5999999999999998e65 < y

   1. Initial program 96.8%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      3. *-commutative N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

      6. lower-/.f64 99.9

         \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{z}}}{3} \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{x - \frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 94.0

         \[\leadsto x - \frac{\color{blue}{\frac{y}{z}}}{3} \]

   9. Applied rewrites 94.0%

      \[\leadsto x - \frac{\color{blue}{\frac{y}{z}}}{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-86} \lor \neg \left(y \leq 5.6 \cdot 10^{+65}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.8e-86) (not (<= y 5.6e+65)))
   (fma -0.3333333333333333 (/ y z) x)
   (fma (/ t (* z y)) 0.3333333333333333 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.8e-86) || !(y <= 5.6e+65)) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.8e-86) || !(y <= 5.6e+65))
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	else
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.8e-86], N[Not[LessEqual[y, 5.6e+65]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999983e-86 or 5.5999999999999998e65 < y

   1. Initial program 96.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \cdot y \]

      3. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \frac{x}{y}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \cdot y \]

      6. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \cdot y \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \cdot y \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot -1\right)} \cdot y \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(-1 \cdot y\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y\right)} \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right) \cdot y} \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      14. +-commutative N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)}\right)\right) \]

      15. distribute-neg-in N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right)} \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}} + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1}\right) \]

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

          \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} - \frac{x}{y} \cdot -1\right)} \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -1.79999999999999983e-86 < y < 5.5999999999999998e65

   1. Initial program 94.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}} \]

      3. associate-/r* N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

      5. lower-/.f64 97.3

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{t}{z \cdot 3}}}{y} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{\color{blue}{z \cdot 3}}}{y} \]

      7. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{\color{blue}{3 \cdot z}}}{y} \]

      8. lower-*.f64 97.3

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{\color{blue}{3 \cdot z}}}{y} \]

   4. Applied rewrites 97.3%

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{3 \cdot z}}{y}} \]

   5. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \frac{t}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot y}}{y} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \frac{t}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}}{y} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot y}{y} \]

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

         \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot y\right)}}{y} \]

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

         \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)}\right)}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right)\right)}{y} \]

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

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot x}\right)}{y} \]

      12. distribute-neg-frac N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot x}{y}\right)\right)} \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\frac{\left(\mathsf{neg}\left(y\right)\right) \cdot x}{\mathsf{neg}\left(y\right)}} \]

      14. associate-*r/ N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{x}{\mathsf{neg}\left(y\right)}} \]

      15. distribute-neg-frac2 N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)} \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{x}{y}\right)\right)} \]

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

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \cdot \frac{x}{y}} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{y} \cdot \frac{x}{y} \]

      19. associate-/l* N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{\frac{y \cdot x}{y}} \]

      20. *-commutative N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \frac{\color{blue}{x \cdot y}}{y} \]

      21. associate-/l* N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{x \cdot \frac{y}{y}} \]

      22. *-inverses N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + x \cdot \color{blue}{1} \]

      23. *-rgt-identity N/A

          \[\leadsto \frac{t}{y \cdot z} \cdot \frac{1}{3} + \color{blue}{x} \]

   7. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-86} \lor \neg \left(y \leq 5.6 \cdot 10^{+65}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-151} \lor \neg \left(y \leq 1.8 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.4e-151) (not (<= y 1.8e-37)))
   (fma -0.3333333333333333 (/ y z) x)
   (* (/ t (* z y)) 0.3333333333333333)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.4e-151) || !(y <= 1.8e-37)) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else {
		tmp = (t / (z * y)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.4e-151) || !(y <= 1.8e-37))
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	else
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.4e-151], N[Not[LessEqual[y, 1.8e-37]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4000000000000003e-151 or 1.80000000000000004e-37 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \cdot y \]

      3. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \frac{x}{y}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \cdot y \]

      6. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \cdot y \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \cdot y \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot -1\right)} \cdot y \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(-1 \cdot y\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y\right)} \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right) \cdot y} \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

      14. +-commutative N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)}\right)\right) \]

      15. distribute-neg-in N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right)} \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}} + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1}\right) \]

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

          \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} - \frac{x}{y} \cdot -1\right)} \]

   5. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -3.4000000000000003e-151 < y < 1.80000000000000004e-37

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z}} \cdot \frac{1}{3} \]

      4. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{z \cdot y}} \cdot \frac{1}{3} \]

      5. lower-*.f64 74.5

         \[\leadsto \frac{t}{\color{blue}{z \cdot y}} \cdot 0.3333333333333333 \]

   5. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\frac{t}{z \cdot y} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-151} \lor \neg \left(y \leq 1.8 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+112} \lor \neg \left(z \leq 5 \cdot 10^{+190}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.45e+112) (not (<= z 5e+190)))
   (* 1.0 x)
   (* (/ y z) -0.3333333333333333)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.45e+112) || !(z <= 5e+190)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y / z) * -0.3333333333333333;
	}
	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)
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) :: tmp
    if ((z <= (-1.45d+112)) .or. (.not. (z <= 5d+190))) then
        tmp = 1.0d0 * x
    else
        tmp = (y / z) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.45e+112) || !(z <= 5e+190)) {
		tmp = 1.0 * x;
	} else {
		tmp = (y / z) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.45e+112) or not (z <= 5e+190):
		tmp = 1.0 * x
	else:
		tmp = (y / z) * -0.3333333333333333
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.45e+112) || !(z <= 5e+190))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(y / z) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.45e+112) || ~((z <= 5e+190)))
		tmp = 1.0 * x;
	else
		tmp = (y / z) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.45e+112], N[Not[LessEqual[z, 5e+190]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4500000000000001e112 or 5.00000000000000036e190 < z

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{1}{3} \cdot \frac{t}{x \cdot \left(y \cdot z\right)}\right) - \frac{1}{3} \cdot \frac{y}{x \cdot z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{3} \cdot \frac{t}{x \cdot \left(y \cdot z\right)}\right) - \frac{1}{3} \cdot \frac{y}{x \cdot z}\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{3} \cdot \frac{t}{x \cdot \left(y \cdot z\right)}\right) - \frac{1}{3} \cdot \frac{y}{x \cdot z}\right) \cdot x} \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{\frac{t}{y} - y}{z}}{x}, 0.3333333333333333, 1\right) \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 60.1%

      \[\leadsto 1 \cdot x \]

   if -1.4500000000000001e112 < z < 5.00000000000000036e190

   1. Initial program 94.6%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 98.3

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 98.3

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{-1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{-1}{3}} \]

      3. lower-/.f64 47.1

         \[\leadsto \color{blue}{\frac{y}{z}} \cdot -0.3333333333333333 \]

   7. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot -0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+112} \lor \neg \left(z \leq 5 \cdot 10^{+190}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot -0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+112} \lor \neg \left(z \leq 5 \cdot 10^{+190}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.45e+112) (not (<= z 5e+190)))
   (* 1.0 x)
   (/ (* -0.3333333333333333 y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.45e+112) || !(z <= 5e+190)) {
		tmp = 1.0 * x;
	} else {
		tmp = (-0.3333333333333333 * y) / z;
	}
	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)
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) :: tmp
    if ((z <= (-1.45d+112)) .or. (.not. (z <= 5d+190))) then
        tmp = 1.0d0 * x
    else
        tmp = ((-0.3333333333333333d0) * y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.45e+112) || !(z <= 5e+190)) {
		tmp = 1.0 * x;
	} else {
		tmp = (-0.3333333333333333 * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.45e+112) or not (z <= 5e+190):
		tmp = 1.0 * x
	else:
		tmp = (-0.3333333333333333 * y) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.45e+112) || !(z <= 5e+190))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(-0.3333333333333333 * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.45e+112) || ~((z <= 5e+190)))
		tmp = 1.0 * x;
	else
		tmp = (-0.3333333333333333 * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.45e+112], N[Not[LessEqual[z, 5e+190]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(N[(-0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4500000000000001e112 or 5.00000000000000036e190 < z

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{1}{3} \cdot \frac{t}{x \cdot \left(y \cdot z\right)}\right) - \frac{1}{3} \cdot \frac{y}{x \cdot z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{3} \cdot \frac{t}{x \cdot \left(y \cdot z\right)}\right) - \frac{1}{3} \cdot \frac{y}{x \cdot z}\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{3} \cdot \frac{t}{x \cdot \left(y \cdot z\right)}\right) - \frac{1}{3} \cdot \frac{y}{x \cdot z}\right) \cdot x} \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{\frac{t}{y} - y}{z}}{x}, 0.3333333333333333, 1\right) \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 60.1%

      \[\leadsto 1 \cdot x \]

   if -1.4500000000000001e112 < z < 5.00000000000000036e190

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\frac{t}{y \cdot z} - \frac{y}{z}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{t}{y \cdot z} - \frac{y}{z}\right) \cdot \frac{1}{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{t}{y \cdot z} - \frac{y}{z}\right) \cdot \frac{1}{3}} \]

      4. associate-/r* N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{t}{y}}{z}} - \frac{y}{z}\right) \cdot \frac{1}{3} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{y} - y}{z}} \cdot \frac{1}{3} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{t}{y} - y}{z}} \cdot \frac{1}{3} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{t}{y} - y}}{z} \cdot \frac{1}{3} \]

      8. lower-/.f64 83.4

         \[\leadsto \frac{\color{blue}{\frac{t}{y}} - y}{z} \cdot 0.3333333333333333 \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\frac{\frac{t}{y} - y}{z} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.4%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{\color{blue}{z}} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{-1}{3} \cdot y}{z} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.1%

       \[\leadsto \frac{-0.3333333333333333 \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+112} \lor \neg \left(z \leq 5 \cdot 10^{+190}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma -0.3333333333333333 (/ y z) x))

C

double code(double x, double y, double z, double t) {
	return fma(-0.3333333333333333, (y / z), x);
}

Julia

function code(x, y, z, t)
	return fma(-0.3333333333333333, Float64(y / z), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y} \]

   2. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \cdot y \]

   3. remove-double-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

   4. mul-1-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \frac{x}{y}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \cdot y \]

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

      \[\leadsto \left(\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \cdot y \]

   6. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \cdot y \]

   7. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \cdot y \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot -1\right)} \cdot y \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(-1 \cdot y\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \cdot y\right)} \]

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

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right) \cdot y} \]

   13. *-commutative N/A

       \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)\right)} \]

   14. +-commutative N/A

       \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)}\right)\right) \]

   15. distribute-neg-in N/A

       \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right)} \]

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

       \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}} + \left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right)\right) \]

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

       \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1}\right) \]

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

       \[\leadsto y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z} - \frac{x}{y} \cdot -1\right)} \]

5. Applied rewrites 65.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

6. Final simplification 65.8%

   \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) \]
7. Add Preprocessing

Alternative 14: 30.8% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t) {
	return 1.0 * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Python

def code(x, y, z, t):
	return 1.0 * x

Julia

function code(x, y, z, t)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{1}{3} \cdot \frac{t}{x \cdot \left(y \cdot z\right)}\right) - \frac{1}{3} \cdot \frac{y}{x \cdot z}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{3} \cdot \frac{t}{x \cdot \left(y \cdot z\right)}\right) - \frac{1}{3} \cdot \frac{y}{x \cdot z}\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{3} \cdot \frac{t}{x \cdot \left(y \cdot z\right)}\right) - \frac{1}{3} \cdot \frac{y}{x \cdot z}\right) \cdot x} \]

5. Applied rewrites 87.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{\frac{t}{y} - y}{z}}{x}, 0.3333333333333333, 1\right) \cdot x} \]

6. Taylor expanded in x around inf

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 28.7%

   \[\leadsto 1 \cdot x \]

9. Final simplification 28.7%

   \[\leadsto 1 \cdot x \]
10. Add Preprocessing

Developer Target 1: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y)))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + ((t / (z * 3.0d0)) / y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y)

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / Float64(z * 3.0)) / y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024360 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y)))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))