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

Percentage Accurate: 95.7% → 98.0%

Time: 4.9s

Alternatives: 19

Speedup: 0.9×

• 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: 95.7% 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: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-111}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z}}{3 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{z \cdot y}, x - \frac{y}{3 \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.5e-111)
   (+ (- x (/ y (* z 3.0))) (/ (/ t z) (* 3.0 y)))
   (fma t (/ 0.3333333333333333 (* z y)) (- x (/ y (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.5e-111) {
		tmp = (x - (y / (z * 3.0))) + ((t / z) / (3.0 * y));
	} else {
		tmp = fma(t, (0.3333333333333333 / (z * y)), (x - (y / (3.0 * z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.5e-111)
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / z) / Float64(3.0 * y)));
	else
		tmp = fma(t, Float64(0.3333333333333333 / Float64(z * y)), Float64(x - Float64(y / Float64(3.0 * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.5e-111], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / z), $MachinePrecision] / N[(3.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.50000000000000004e-111

   1. Initial program 95.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 96.1

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

   3. Applied rewrites 96.1%

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

   if 1.50000000000000004e-111 < t

   1. Initial program 95.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. 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. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 95.7

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

   3. Applied rewrites 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. mult-flip-rev N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 95.5%

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

Alternative 2: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{z \cdot y}, x - \frac{y}{3 \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1e-111)
   (fma (/ t z) (/ 0.3333333333333333 y) (fma -0.3333333333333333 (/ y z) x))
   (fma t (/ 0.3333333333333333 (* z y)) (- x (/ y (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1e-111) {
		tmp = fma((t / z), (0.3333333333333333 / y), fma(-0.3333333333333333, (y / z), x));
	} else {
		tmp = fma(t, (0.3333333333333333 / (z * y)), (x - (y / (3.0 * z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1e-111)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), fma(-0.3333333333333333, Float64(y / z), x));
	else
		tmp = fma(t, Float64(0.3333333333333333 / Float64(z * y)), Float64(x - Float64(y / Float64(3.0 * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1e-111], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000009e-111

   1. Initial program 95.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. mult-flip N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. associate-*l* N/A

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

      16. mult-flip N/A

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

      17. *-commutative N/A

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

   3. Applied rewrites 96.0%

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

   if 1.00000000000000009e-111 < t

   1. Initial program 95.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. 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. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 95.7

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

   3. Applied rewrites 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. mult-flip-rev N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 95.5%

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

Alternative 3: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;t \leq 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z \cdot y}, t, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= t 1e-111)
     (fma (/ t z) (/ 0.3333333333333333 y) t_1)
     (fma (/ 0.3333333333333333 (* z y)) t t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (t <= 1e-111) {
		tmp = fma((t / z), (0.3333333333333333 / y), t_1);
	} else {
		tmp = fma((0.3333333333333333 / (z * y)), t, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (t <= 1e-111)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), t_1);
	else
		tmp = fma(Float64(0.3333333333333333 / Float64(z * y)), t, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, 1e-111], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] * t + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000009e-111

   1. Initial program 95.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. mult-flip N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. associate-*l* N/A

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

      16. mult-flip N/A

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

      17. *-commutative N/A

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

   3. Applied rewrites 96.0%

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

   if 1.00000000000000009e-111 < t

   1. Initial program 95.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. 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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-/r* N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. *-commutative N/A

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

      16. mult-flip-rev N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 95.5%

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

Alternative 4: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\left(\frac{t}{y} - y\right) \cdot -0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{z}, 0.3333333333333333, y \cdot x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (* (- (/ t y) y) -0.3333333333333333) z))))
   (if (<= y -4.1e-29)
     t_1
     (if (<= y 7.2e-78) (/ (fma (/ t z) 0.3333333333333333 (* y x)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((((t / y) - y) * -0.3333333333333333) / z);
	double tmp;
	if (y <= -4.1e-29) {
		tmp = t_1;
	} else if (y <= 7.2e-78) {
		tmp = fma((t / z), 0.3333333333333333, (y * x)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(Float64(Float64(t / y) - y) * -0.3333333333333333) / z))
	tmp = 0.0
	if (y <= -4.1e-29)
		tmp = t_1;
	elseif (y <= 7.2e-78)
		tmp = Float64(fma(Float64(t / z), 0.3333333333333333, Float64(y * x)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.1e-29], t$95$1, If[LessEqual[y, 7.2e-78], N[(N[(N[(t / z), $MachinePrecision] * 0.3333333333333333 + N[(y * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0999999999999998e-29 or 7.2000000000000005e-78 < y

   1. Initial program 95.7%

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

   3. Applied rewrites 95.8%

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

   if -4.0999999999999998e-29 < y < 7.2000000000000005e-78

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 60.8

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

   4. Applied rewrites 60.8%

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

Alternative 5: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{z}, 0.3333333333333333, y \cdot x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ (- (/ t y) y) z) 0.3333333333333333 x)))
   (if (<= y -4.1e-29)
     t_1
     (if (<= y 5.5e-82) (/ (fma (/ t z) 0.3333333333333333 (* y x)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((((t / y) - y) / z), 0.3333333333333333, x);
	double tmp;
	if (y <= -4.1e-29) {
		tmp = t_1;
	} else if (y <= 5.5e-82) {
		tmp = fma((t / z), 0.3333333333333333, (y * x)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(Float64(Float64(t / y) - y) / z), 0.3333333333333333, x)
	tmp = 0.0
	if (y <= -4.1e-29)
		tmp = t_1;
	elseif (y <= 5.5e-82)
		tmp = Float64(fma(Float64(t / z), 0.3333333333333333, Float64(y * x)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision]}, If[LessEqual[y, -4.1e-29], t$95$1, If[LessEqual[y, 5.5e-82], N[(N[(N[(t / z), $MachinePrecision] * 0.3333333333333333 + N[(y * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0999999999999998e-29 or 5.4999999999999998e-82 < y

   1. Initial program 95.7%

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

   3. Applied rewrites 95.7%

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

   if -4.0999999999999998e-29 < y < 5.4999999999999998e-82

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 60.8

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

   4. Applied rewrites 60.8%

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

Alternative 6: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;x - \frac{\frac{y}{z} - \frac{t}{z \cdot y}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, \left(\frac{t}{y} - y\right) \cdot 0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))) 2e+294)
   (- x (/ (- (/ y z) (/ t (* z y))) 3.0))
   (fma (/ 1.0 z) (* (- (/ t y) y) 0.3333333333333333) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))) <= 2e+294) {
		tmp = x - (((y / z) - (t / (z * y))) / 3.0);
	} else {
		tmp = fma((1.0 / z), (((t / y) - y) * 0.3333333333333333), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y))) <= 2e+294)
		tmp = Float64(x - Float64(Float64(Float64(y / z) - Float64(t / Float64(z * y))) / 3.0));
	else
		tmp = fma(Float64(1.0 / z), Float64(Float64(Float64(t / y) - y) * 0.3333333333333333), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+294], N[(x - N[(N[(N[(y / z), $MachinePrecision] - N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + x), $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))) < 2.00000000000000013e294

   1. Initial program 95.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. 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. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 95.7

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

   3. Applied rewrites 95.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-/.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if 2.00000000000000013e294 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 95.7%

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

   3. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{z}, \left(\frac{t}{y} - y\right) \cdot 0.3333333333333333, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 95.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. 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. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. mult-flip N/A

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

   10. associate-/r* N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. associate-/r* N/A

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

   15. *-commutative N/A

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

   16. mult-flip-rev N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

3. Applied rewrites 95.5%

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

Alternative 8: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+26}:\\ \;\;\;\;\left(\frac{x}{y} - \frac{0.3333333333333333}{z}\right) \cdot y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{t}{y} - y}{z} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{z}, -0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.22e+26)
   (* (- (/ x y) (/ 0.3333333333333333 z)) y)
   (if (<= y 8.5e-49)
     (fma (/ t (* z y)) 0.3333333333333333 x)
     (if (<= y 1.16e+80)
       (* (/ (- (/ t y) y) z) 0.3333333333333333)
       (fma (* y (/ 1.0 z)) -0.3333333333333333 x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.22e+26) {
		tmp = ((x / y) - (0.3333333333333333 / z)) * y;
	} else if (y <= 8.5e-49) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else if (y <= 1.16e+80) {
		tmp = (((t / y) - y) / z) * 0.3333333333333333;
	} else {
		tmp = fma((y * (1.0 / z)), -0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.22e+26)
		tmp = Float64(Float64(Float64(x / y) - Float64(0.3333333333333333 / z)) * y);
	elseif (y <= 8.5e-49)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	elseif (y <= 1.16e+80)
		tmp = Float64(Float64(Float64(Float64(t / y) - y) / z) * 0.3333333333333333);
	else
		tmp = fma(Float64(y * Float64(1.0 / z)), -0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.22e+26], N[(N[(N[(x / y), $MachinePrecision] - N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 8.5e-49], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], If[LessEqual[y, 1.16e+80], N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(y * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.2200000000000001e26

   1. Initial program 95.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 61.2

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

   4. Applied rewrites 61.2%

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

   if -1.2200000000000001e26 < y < 8.50000000000000069e-49

   1. Initial program 95.7%

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

   3. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add-rev N/A

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

      6. associate-/r* N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. mult-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. mul-1-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 8.50000000000000069e-49 < y < 1.15999999999999997e80

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 67.8

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

   4. Applied rewrites 67.8%

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

   if 1.15999999999999997e80 < y

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 64.5

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

   7. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \frac{1}{z}, \frac{-1}{3}, x\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \frac{1}{z}, \frac{-1}{3}, x\right) \]

      4. lower-/.f64 64.5

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

   9. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(y \cdot \frac{1}{z}, -0.3333333333333333, x\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Applied rewrites 95.7%

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

Alternative 10: 90.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+26}:\\ \;\;\;\;\left(\frac{x}{y} - \frac{0.3333333333333333}{z}\right) \cdot y\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.22e+26)
   (* (- (/ x y) (/ 0.3333333333333333 z)) y)
   (if (<= y 2.05e+30)
     (fma (/ t (* z y)) 0.3333333333333333 x)
     (fma (/ y z) -0.3333333333333333 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.22e+26) {
		tmp = ((x / y) - (0.3333333333333333 / z)) * y;
	} else if (y <= 2.05e+30) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = fma((y / z), -0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.22e+26)
		tmp = Float64(Float64(Float64(x / y) - Float64(0.3333333333333333 / z)) * y);
	elseif (y <= 2.05e+30)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = fma(Float64(y / z), -0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.22e+26], N[(N[(N[(x / y), $MachinePrecision] - N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.05e+30], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2200000000000001e26

   1. Initial program 95.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 61.2

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

   4. Applied rewrites 61.2%

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

   if -1.2200000000000001e26 < y < 2.05000000000000003e30

   1. Initial program 95.7%

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

   3. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add-rev N/A

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

      6. associate-/r* N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. mult-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. mul-1-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 2.05000000000000003e30 < y

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

Alternative 11: 90.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, -0.3333333333333333 \cdot y, x\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.22e+26)
   (fma (/ 1.0 z) (* -0.3333333333333333 y) x)
   (if (<= y 2.05e+30)
     (fma (/ t (* z y)) 0.3333333333333333 x)
     (fma (/ y z) -0.3333333333333333 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.22e+26) {
		tmp = fma((1.0 / z), (-0.3333333333333333 * y), x);
	} else if (y <= 2.05e+30) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = fma((y / z), -0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.22e+26)
		tmp = fma(Float64(1.0 / z), Float64(-0.3333333333333333 * y), x);
	elseif (y <= 2.05e+30)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = fma(Float64(y / z), -0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.22e+26], N[(N[(1.0 / z), $MachinePrecision] * N[(-0.3333333333333333 * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.05e+30], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2200000000000001e26

   1. Initial program 95.7%

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -1.2200000000000001e26 < y < 2.05000000000000003e30

   1. Initial program 95.7%

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

   3. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add-rev N/A

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

      6. associate-/r* N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. mult-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. mul-1-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 2.05000000000000003e30 < y

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

Alternative 12: 88.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{if}\;y \leq -1.22 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y z) -0.3333333333333333 x)))
   (if (<= y -1.22e+26)
     t_1
     (if (<= y 2.05e+30) (fma (/ t (* z y)) 0.3333333333333333 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / z), -0.3333333333333333, x);
	double tmp;
	if (y <= -1.22e+26) {
		tmp = t_1;
	} else if (y <= 2.05e+30) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / z), -0.3333333333333333, x)
	tmp = 0.0
	if (y <= -1.22e+26)
		tmp = t_1;
	elseif (y <= 2.05e+30)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]}, If[LessEqual[y, -1.22e+26], t$95$1, If[LessEqual[y, 2.05e+30], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2200000000000001e26 or 2.05000000000000003e30 < y

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

   if -1.2200000000000001e26 < y < 2.05000000000000003e30

   1. Initial program 95.7%

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

   3. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add-rev N/A

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

      6. associate-/r* N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. mult-flip N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. mul-1-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

Alternative 13: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y z) -0.3333333333333333 x)))
   (if (<= y -6.6e-102)
     t_1
     (if (<= y 2.2e-73) (* (/ (/ t z) y) 0.3333333333333333) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / z), -0.3333333333333333, x);
	double tmp;
	if (y <= -6.6e-102) {
		tmp = t_1;
	} else if (y <= 2.2e-73) {
		tmp = ((t / z) / y) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / z), -0.3333333333333333, x)
	tmp = 0.0
	if (y <= -6.6e-102)
		tmp = t_1;
	elseif (y <= 2.2e-73)
		tmp = Float64(Float64(Float64(t / z) / y) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]}, If[LessEqual[y, -6.6e-102], t$95$1, If[LessEqual[y, 2.2e-73], N[(N[(N[(t / z), $MachinePrecision] / y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.6e-102 or 2.2e-73 < y

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

   if -6.6e-102 < y < 2.2e-73

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 37.8

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

   6. Applied rewrites 37.8%

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

Alternative 14: 77.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y z) -0.3333333333333333 x)))
   (if (<= y -6.6e-102) t_1 (if (<= y 2.2e-73) (/ t (* (* 3.0 z) y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / z), -0.3333333333333333, x);
	double tmp;
	if (y <= -6.6e-102) {
		tmp = t_1;
	} else if (y <= 2.2e-73) {
		tmp = t / ((3.0 * z) * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / z), -0.3333333333333333, x)
	tmp = 0.0
	if (y <= -6.6e-102)
		tmp = t_1;
	elseif (y <= 2.2e-73)
		tmp = Float64(t / Float64(Float64(3.0 * z) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]}, If[LessEqual[y, -6.6e-102], t$95$1, If[LessEqual[y, 2.2e-73], N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.6e-102 or 2.2e-73 < y

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

   if -6.6e-102 < y < 2.2e-73

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. mult-flip-rev N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. times-frac N/A

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

      17. metadata-eval N/A

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

      18. times-frac N/A

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

      19. metadata-eval N/A

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

      20. mult-flip N/A

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

      21. lower-/.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 34.7

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

   6. Applied rewrites 34.7%

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

Alternative 15: 77.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y z) -0.3333333333333333 x)))
   (if (<= y -6.6e-102)
     t_1
     (if (<= y 2.2e-73) (* (/ t (* z y)) 0.3333333333333333) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / z), -0.3333333333333333, x);
	double tmp;
	if (y <= -6.6e-102) {
		tmp = t_1;
	} else if (y <= 2.2e-73) {
		tmp = (t / (z * y)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / z), -0.3333333333333333, x)
	tmp = 0.0
	if (y <= -6.6e-102)
		tmp = t_1;
	elseif (y <= 2.2e-73)
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]}, If[LessEqual[y, -6.6e-102], t$95$1, If[LessEqual[y, 2.2e-73], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.6e-102 or 2.2e-73 < y

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

   if -6.6e-102 < y < 2.2e-73

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

Alternative 16: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.3333333333333333}{y \cdot z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y z) -0.3333333333333333 x)))
   (if (<= y -6.6e-102)
     t_1
     (if (<= y 2.2e-73) (* (/ 0.3333333333333333 (* y z)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / z), -0.3333333333333333, x);
	double tmp;
	if (y <= -6.6e-102) {
		tmp = t_1;
	} else if (y <= 2.2e-73) {
		tmp = (0.3333333333333333 / (y * z)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / z), -0.3333333333333333, x)
	tmp = 0.0
	if (y <= -6.6e-102)
		tmp = t_1;
	elseif (y <= 2.2e-73)
		tmp = Float64(Float64(0.3333333333333333 / Float64(y * z)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]}, If[LessEqual[y, -6.6e-102], t$95$1, If[LessEqual[y, 2.2e-73], N[(N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.6e-102 or 2.2e-73 < y

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

   if -6.6e-102 < y < 2.2e-73

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. associate-*l/ N/A

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

      10. mult-flip N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 34.5

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

   6. Applied rewrites 34.5%

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

Alternative 17: 64.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 34.6

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

4. Applied rewrites 34.6%

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

5. Taylor expanded in t around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

      \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

   5. lower-fma.f64 N/A

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

   6. lift-/.f64 64.5

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

7. Applied rewrites 64.5%

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

Alternative 18: 36.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333 \cdot y}{z} \end{array} \]

FPCore

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

C

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

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 = ((-0.3333333333333333d0) * y) / z
end function

Java

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

Python

def code(x, y, z, t):
	return (-0.3333333333333333 * y) / z

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-0.3333333333333333 * y) / z;
end

Wolfram

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

Derivation

1. Initial program 95.7%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 34.6

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

4. Applied rewrites 34.6%

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

5. Taylor expanded in t around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

      \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

   5. lower-fma.f64 N/A

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

   6. lift-/.f64 64.5

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

7. Applied rewrites 64.5%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-/.f64 36.4

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

10. Applied rewrites 36.4%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-*r/ N/A

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

    5. lower-/.f64 N/A

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

    6. lower-*.f64 36.4

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

12. Applied rewrites 36.4%

    \[\leadsto \frac{-0.3333333333333333 \cdot y}{z} \]
13. Add Preprocessing

Alternative 19: 36.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{y}{z} \cdot -0.3333333333333333 \end{array} \]

FPCore

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

C

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

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 = (y / z) * (-0.3333333333333333d0)
end function

Java

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

Python

def code(x, y, z, t):
	return (y / z) * -0.3333333333333333

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y / z) * -0.3333333333333333;
end

Wolfram

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

Derivation

1. Initial program 95.7%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 34.6

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

4. Applied rewrites 34.6%

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

5. Taylor expanded in t around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

      \[\leadsto \frac{y}{z} \cdot \frac{-1}{3} + x \]

   5. lower-fma.f64 N/A

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

   6. lift-/.f64 64.5

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

7. Applied rewrites 64.5%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-/.f64 36.4

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

10. Applied rewrites 36.4%

    \[\leadsto \frac{y}{z} \cdot \color{blue}{-0.3333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64
  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))