Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B

Percentage Accurate: 95.9% → 95.9%

Time: 3.4s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{-1 \cdot x}{-\mathsf{fma}\left(-1 \cdot t, z, y\right)} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (-1.0 * x) / -fma((-1.0 * t), z, y);
}

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(-1.0 * x), $MachinePrecision] / (-N[(N[(-1.0 * t), $MachinePrecision] * z + y), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 95.9%

   \[\frac{x}{y - z \cdot t} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. frac-2neg N/A

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

   5. mul-1-neg N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. mul-1-neg N/A

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

   12. associate-*r* N/A

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

   13. +-commutative N/A

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

   14. associate-*r* N/A

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

   15. mul-1-neg N/A

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

   16. lower-fma.f64 N/A

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

   17. mul-1-neg N/A

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

   18. lower-*.f64 95.9

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

3. Applied rewrites 95.9%

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

Alternative 2: 95.9% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

   \[\frac{x}{y - z \cdot t} \]
2. Add Preprocessing

Alternative 3: 40.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\left(y \cdot z\right) \cdot t}, -1, \frac{x}{{\left(\left(z \cdot t\right) \cdot -1\right)}^{2}} \cdot -1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \frac{x}{t}, \frac{-1 \cdot \left(y \cdot x\right)}{{\left(t \cdot z\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.18e-36)
   (*
    (fma (/ x (* (* y z) t)) -1.0 (* (/ x (pow (* (* z t) -1.0) 2.0)) -1.0))
    y)
   (fma (/ -1.0 z) (/ x t) (/ (* -1.0 (* y x)) (pow (* t z) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.18e-36) {
		tmp = fma((x / ((y * z) * t)), -1.0, ((x / pow(((z * t) * -1.0), 2.0)) * -1.0)) * y;
	} else {
		tmp = fma((-1.0 / z), (x / t), ((-1.0 * (y * x)) / pow((t * z), 2.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.18e-36)
		tmp = Float64(fma(Float64(x / Float64(Float64(y * z) * t)), -1.0, Float64(Float64(x / (Float64(Float64(z * t) * -1.0) ^ 2.0)) * -1.0)) * y);
	else
		tmp = fma(Float64(-1.0 / z), Float64(x / t), Float64(Float64(-1.0 * Float64(y * x)) / (Float64(t * z) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.18e-36], N[(N[(N[(x / N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[Power[N[(N[(z * t), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * N[(x / t), $MachinePrecision] + N[(N[(-1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t * z), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1799999999999999e-36

   1. Initial program 97.1%

      \[\frac{x}{y - z \cdot t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{\color{blue}{t}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{{z}^{2} \cdot t} + -1 \cdot \frac{x}{z}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{-1}{{z}^{2}} \cdot \frac{x \cdot y}{t} + -1 \cdot \frac{x}{z}}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 42.2

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

   4. Applied rewrites 42.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

   7. Applied rewrites 40.1%

      \[\leadsto \mathsf{fma}\left(\frac{x}{\left(y \cdot z\right) \cdot t}, -1, \frac{x}{{\left(\left(z \cdot t\right) \cdot -1\right)}^{2}} \cdot -1\right) \cdot \color{blue}{y} \]

   if 1.1799999999999999e-36 < x

   1. Initial program 92.9%

      \[\frac{x}{y - z \cdot t} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot z} + -1 \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{t \cdot z} + \color{blue}{-1} \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1 \cdot x}{z \cdot t} + -1 \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}} \]

      3. times-frac N/A

         \[\leadsto \frac{-1}{z} \cdot \frac{x}{t} + \color{blue}{-1} \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \color{blue}{\frac{x}{t}}, -1 \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{\color{blue}{x}}{t}, -1 \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{x}{\color{blue}{t}}, -1 \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{x}{t}, \frac{-1 \cdot \left(x \cdot y\right)}{{t}^{2} \cdot {z}^{2}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{x}{t}, \frac{-1 \cdot \left(x \cdot y\right)}{{t}^{2} \cdot {z}^{2}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{x}{t}, \frac{-1 \cdot \left(x \cdot y\right)}{{t}^{2} \cdot {z}^{2}}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{x}{t}, \frac{-1 \cdot \left(y \cdot x\right)}{{t}^{2} \cdot {z}^{2}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{x}{t}, \frac{-1 \cdot \left(y \cdot x\right)}{{t}^{2} \cdot {z}^{2}}\right) \]

      12. pow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{x}{t}, \frac{-1 \cdot \left(y \cdot x\right)}{{\left(t \cdot z\right)}^{2}}\right) \]

      13. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{x}{t}, \frac{-1 \cdot \left(y \cdot x\right)}{{\left(t \cdot z\right)}^{2}}\right) \]

      14. lower-*.f64 39.3

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \frac{x}{t}, \frac{-1 \cdot \left(y \cdot x\right)}{{\left(t \cdot z\right)}^{2}}\right) \]

   4. Applied rewrites 39.3%

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

Alternative 4: 40.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-176}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{z \cdot z}, \frac{y \cdot x}{t}, \frac{x}{z} \cdot -1\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\left(y \cdot z\right) \cdot t}, -1, \frac{x}{{\left(\left(z \cdot t\right) \cdot -1\right)}^{2}} \cdot -1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.4e-176)
   (/ (fma (/ -1.0 (* z z)) (/ (* y x) t) (* (/ x z) -1.0)) t)
   (*
    (fma (/ x (* (* y z) t)) -1.0 (* (/ x (pow (* (* z t) -1.0) 2.0)) -1.0))
    y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.4e-176) {
		tmp = fma((-1.0 / (z * z)), ((y * x) / t), ((x / z) * -1.0)) / t;
	} else {
		tmp = fma((x / ((y * z) * t)), -1.0, ((x / pow(((z * t) * -1.0), 2.0)) * -1.0)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.4e-176)
		tmp = Float64(fma(Float64(-1.0 / Float64(z * z)), Float64(Float64(y * x) / t), Float64(Float64(x / z) * -1.0)) / t);
	else
		tmp = Float64(fma(Float64(x / Float64(Float64(y * z) * t)), -1.0, Float64(Float64(x / (Float64(Float64(z * t) * -1.0) ^ 2.0)) * -1.0)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.4e-176], N[(N[(N[(-1.0 / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision] + N[(N[(x / z), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(N[(x / N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[Power[N[(N[(z * t), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.40000000000000006e-176

   1. Initial program 95.9%

      \[\frac{x}{y - z \cdot t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{\color{blue}{t}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{{z}^{2} \cdot t} + -1 \cdot \frac{x}{z}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{-1}{{z}^{2}} \cdot \frac{x \cdot y}{t} + -1 \cdot \frac{x}{z}}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 45.4

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

   4. Applied rewrites 45.4%

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

   if 2.40000000000000006e-176 < y

   1. Initial program 96.0%

      \[\frac{x}{y - z \cdot t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{\color{blue}{t}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{{z}^{2} \cdot t} + -1 \cdot \frac{x}{z}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{-1}{{z}^{2}} \cdot \frac{x \cdot y}{t} + -1 \cdot \frac{x}{z}}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 33.3

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

   4. Applied rewrites 33.3%

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

   5. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

   7. Applied rewrites 34.0%

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

Alternative 5: 39.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{\left(z \cdot z\right) \cdot z} \cdot {\left(\frac{y}{t} \cdot -1\right)}^{2}, -1, \frac{x}{z} \cdot -1\right) - \frac{x}{z \cdot z} \cdot \frac{y}{t}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\left(y \cdot z\right) \cdot t}, -1, \frac{x}{{\left(\left(z \cdot t\right) \cdot -1\right)}^{2}} \cdot -1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{y}{z} \cdot -1\right)}^{2} \cdot \frac{x}{\left(t \cdot t\right) \cdot t}, -1, \frac{x}{t} \cdot -1\right) - \frac{y}{t \cdot t} \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+170)
   (/
    (-
     (fma
      (* (/ x (* (* z z) z)) (pow (* (/ y t) -1.0) 2.0))
      -1.0
      (* (/ x z) -1.0))
     (* (/ x (* z z)) (/ y t)))
    t)
   (if (<= (* z t) 5e+216)
     (*
      (fma (/ x (* (* y z) t)) -1.0 (* (/ x (pow (* (* z t) -1.0) 2.0)) -1.0))
      y)
     (/
      (-
       (fma
        (* (pow (* (/ y z) -1.0) 2.0) (/ x (* (* t t) t)))
        -1.0
        (* (/ x t) -1.0))
       (* (/ y (* t t)) (/ x z)))
      z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+170) {
		tmp = (fma(((x / ((z * z) * z)) * pow(((y / t) * -1.0), 2.0)), -1.0, ((x / z) * -1.0)) - ((x / (z * z)) * (y / t))) / t;
	} else if ((z * t) <= 5e+216) {
		tmp = fma((x / ((y * z) * t)), -1.0, ((x / pow(((z * t) * -1.0), 2.0)) * -1.0)) * y;
	} else {
		tmp = (fma((pow(((y / z) * -1.0), 2.0) * (x / ((t * t) * t))), -1.0, ((x / t) * -1.0)) - ((y / (t * t)) * (x / z))) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+170)
		tmp = Float64(Float64(fma(Float64(Float64(x / Float64(Float64(z * z) * z)) * (Float64(Float64(y / t) * -1.0) ^ 2.0)), -1.0, Float64(Float64(x / z) * -1.0)) - Float64(Float64(x / Float64(z * z)) * Float64(y / t))) / t);
	elseif (Float64(z * t) <= 5e+216)
		tmp = Float64(fma(Float64(x / Float64(Float64(y * z) * t)), -1.0, Float64(Float64(x / (Float64(Float64(z * t) * -1.0) ^ 2.0)) * -1.0)) * y);
	else
		tmp = Float64(Float64(fma(Float64((Float64(Float64(y / z) * -1.0) ^ 2.0) * Float64(x / Float64(Float64(t * t) * t))), -1.0, Float64(Float64(x / t) * -1.0)) - Float64(Float64(y / Float64(t * t)) * Float64(x / z))) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+170], N[(N[(N[(N[(N[(x / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(y / t), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / z), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+216], N[(N[(N[(x / N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[Power[N[(N[(z * t), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[Power[N[(N[(y / z), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision] * N[(x / N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / t), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000007e170

   1. Initial program 84.7%

      \[\frac{x}{y - z \cdot t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot {y}^{2}}{{t}^{2} \cdot {z}^{3}}\right) - \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]

   4. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x}{\left(z \cdot z\right) \cdot z} \cdot {\left(\frac{y}{t} \cdot -1\right)}^{2}, -1, \frac{x}{z} \cdot -1\right) - \frac{x}{z \cdot z} \cdot \frac{y}{t}}{t}} \]

   if -2.00000000000000007e170 < (*.f64 z t) < 4.9999999999999998e216

   1. Initial program 99.9%

      \[\frac{x}{y - z \cdot t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{\color{blue}{t}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{{z}^{2} \cdot t} + -1 \cdot \frac{x}{z}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{-1}{{z}^{2}} \cdot \frac{x \cdot y}{t} + -1 \cdot \frac{x}{z}}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 27.0

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

   7. Applied rewrites 24.8%

      \[\leadsto \mathsf{fma}\left(\frac{x}{\left(y \cdot z\right) \cdot t}, -1, \frac{x}{{\left(\left(z \cdot t\right) \cdot -1\right)}^{2}} \cdot -1\right) \cdot \color{blue}{y} \]

   if 4.9999999999999998e216 < (*.f64 z t)

   1. Initial program 80.9%

      \[\frac{x}{y - z \cdot t} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{x}{t} + -1 \cdot \frac{x \cdot {y}^{2}}{{t}^{3} \cdot {z}^{2}}\right) - \frac{x \cdot y}{{t}^{2} \cdot z}}{z}} \]

   4. Applied rewrites 93.5%

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

Alternative 6: 38.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{\left(z \cdot z\right) \cdot z} \cdot {\left(\frac{y}{t} \cdot -1\right)}^{2}, -1, \frac{x}{z} \cdot -1\right) - \frac{x}{z \cdot z} \cdot \frac{y}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\left(y \cdot z\right) \cdot t}, -1, \frac{x}{{\left(\left(z \cdot t\right) \cdot -1\right)}^{2}} \cdot -1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+170)
   (/
    (-
     (fma
      (* (/ x (* (* z z) z)) (pow (* (/ y t) -1.0) 2.0))
      -1.0
      (* (/ x z) -1.0))
     (* (/ x (* z z)) (/ y t)))
    t)
   (*
    (fma (/ x (* (* y z) t)) -1.0 (* (/ x (pow (* (* z t) -1.0) 2.0)) -1.0))
    y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+170) {
		tmp = (fma(((x / ((z * z) * z)) * pow(((y / t) * -1.0), 2.0)), -1.0, ((x / z) * -1.0)) - ((x / (z * z)) * (y / t))) / t;
	} else {
		tmp = fma((x / ((y * z) * t)), -1.0, ((x / pow(((z * t) * -1.0), 2.0)) * -1.0)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+170)
		tmp = Float64(Float64(fma(Float64(Float64(x / Float64(Float64(z * z) * z)) * (Float64(Float64(y / t) * -1.0) ^ 2.0)), -1.0, Float64(Float64(x / z) * -1.0)) - Float64(Float64(x / Float64(z * z)) * Float64(y / t))) / t);
	else
		tmp = Float64(fma(Float64(x / Float64(Float64(y * z) * t)), -1.0, Float64(Float64(x / (Float64(Float64(z * t) * -1.0) ^ 2.0)) * -1.0)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+170], N[(N[(N[(N[(N[(x / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(y / t), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / z), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(N[(x / N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[Power[N[(N[(z * t), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000007e170

   1. Initial program 84.7%

      \[\frac{x}{y - z \cdot t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot {y}^{2}}{{t}^{2} \cdot {z}^{3}}\right) - \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]

   4. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x}{\left(z \cdot z\right) \cdot z} \cdot {\left(\frac{y}{t} \cdot -1\right)}^{2}, -1, \frac{x}{z} \cdot -1\right) - \frac{x}{z \cdot z} \cdot \frac{y}{t}}{t}} \]

   if -2.00000000000000007e170 < (*.f64 z t)

   1. Initial program 97.5%

      \[\frac{x}{y - z \cdot t} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{\color{blue}{t}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{{z}^{2} \cdot t} + -1 \cdot \frac{x}{z}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{-1}{{z}^{2}} \cdot \frac{x \cdot y}{t} + -1 \cdot \frac{x}{z}}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 34.5

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

   4. Applied rewrites 34.5%

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

   5. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

   7. Applied rewrites 31.4%

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

Alternative 7: 37.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{\left(y \cdot z\right) \cdot t}, -1, \frac{x}{{\left(\left(z \cdot t\right) \cdot -1\right)}^{2}} \cdot -1\right) \cdot y \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (fma (/ x (* (* y z) t)) -1.0 (* (/ x (pow (* (* z t) -1.0) 2.0)) -1.0))
  y))

C

double code(double x, double y, double z, double t) {
	return fma((x / ((y * z) * t)), -1.0, ((x / pow(((z * t) * -1.0), 2.0)) * -1.0)) * y;
}

Julia

function code(x, y, z, t)
	return Float64(fma(Float64(x / Float64(Float64(y * z) * t)), -1.0, Float64(Float64(x / (Float64(Float64(z * t) * -1.0) ^ 2.0)) * -1.0)) * y)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x / N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[Power[N[(N[(z * t), $MachinePrecision] * -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 95.9%

   \[\frac{x}{y - z \cdot t} \]

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{\color{blue}{t}} \]

   2. +-commutative N/A

      \[\leadsto \frac{-1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{t \cdot {z}^{2}} + -1 \cdot \frac{x}{z}}{t} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot y\right)}{{z}^{2} \cdot t} + -1 \cdot \frac{x}{z}}{t} \]

   5. times-frac N/A

      \[\leadsto \frac{\frac{-1}{{z}^{2}} \cdot \frac{x \cdot y}{t} + -1 \cdot \frac{x}{z}}{t} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{{z}^{2}}, \frac{x \cdot y}{t}, -1 \cdot \frac{x}{z}\right)}{t} \]

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 40.7

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

4. Applied rewrites 40.7%

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

5. Taylor expanded in y around inf

   \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot \frac{x}{t \cdot \left(y \cdot z\right)} + -1 \cdot \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

7. Applied rewrites 37.3%

   \[\leadsto \mathsf{fma}\left(\frac{x}{\left(y \cdot z\right) \cdot t}, -1, \frac{x}{{\left(\left(z \cdot t\right) \cdot -1\right)}^{2}} \cdot -1\right) \cdot \color{blue}{y} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64
  (/ x (- y (* z t))))