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

Percentage Accurate: 85.4% → 95.0%

Time: 5.4s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1} - y \cdot \frac{z}{t\_1}\\ t_3 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-150}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{a} - t \cdot \frac{y}{a \cdot a}}{z}, -1, \frac{y}{a}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a \cdot z}, -1, \frac{y}{a}\right) + \frac{t}{a \cdot a} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z)))
        (t_2 (- (/ x t_1) (* y (/ z t_1))))
        (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 -5e-150)
     t_2
     (if (<= t_3 -5e-321)
       t_3
       (if (<= t_3 0.0)
         (fma (/ (- (/ x a) (* t (/ y (* a a)))) z) -1.0 (/ y a))
         (if (<= t_3 INFINITY)
           t_2
           (+ (fma (/ x (* a z)) -1.0 (/ y a)) (* (/ t (* a a)) (/ y z)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y * (z / t_1));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -5e-150) {
		tmp = t_2;
	} else if (t_3 <= -5e-321) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = fma((((x / a) - (t * (y / (a * a)))) / z), -1.0, (y / a));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma((x / (a * z)), -1.0, (y / a)) + ((t / (a * a)) * (y / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x / t_1) - Float64(y * Float64(z / t_1)))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= -5e-150)
		tmp = t_2;
	elseif (t_3 <= -5e-321)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = fma(Float64(Float64(Float64(x / a) - Float64(t * Float64(y / Float64(a * a)))) / z), -1.0, Float64(y / a));
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(fma(Float64(x / Float64(a * z)), -1.0, Float64(y / a)) + Float64(Float64(t / Float64(a * a)) * Float64(y / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-150], t$95$2, If[LessEqual[t$95$3, -5e-321], t$95$3, If[LessEqual[t$95$3, 0.0], N[(N[(N[(N[(x / a), $MachinePrecision] - N[(t * N[(y / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -1.0 + N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(N[(N[(x / N[(a * z), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(y / a), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.9999999999999999e-150 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 93.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if -4.9999999999999999e-150 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99994e-321

   1. Initial program 99.6%

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

   if -4.99994e-321 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 49.8%

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

   3. Taylor expanded in z around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -5 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{z}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{a} - t \cdot \frac{y}{a \cdot a}}{z}, -1, \frac{y}{a}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a \cdot z}, -1, \frac{y}{a}\right) + \frac{t}{a \cdot a} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.5% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 90.8

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

   4. Applied rewrites 90.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a \cdot z}, -1, \frac{y}{a}\right) + \frac{t}{a \cdot a} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-29} \lor \neg \left(a \leq 4100000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{a} - t \cdot \frac{y}{a \cdot a}}{z}, -1, \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(a, \frac{z}{t}, 1\right)}{t}, \frac{y \cdot z}{-t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.5e-29) (not (<= a 4100000.0)))
   (fma (/ (- (/ x a) (* t (/ y (* a a)))) z) -1.0 (/ y a))
   (fma x (/ (fma a (/ z t) 1.0) t) (/ (* y z) (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.5e-29) || !(a <= 4100000.0)) {
		tmp = fma((((x / a) - (t * (y / (a * a)))) / z), -1.0, (y / a));
	} else {
		tmp = fma(x, (fma(a, (z / t), 1.0) / t), ((y * z) / -t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.5e-29) || !(a <= 4100000.0))
		tmp = fma(Float64(Float64(Float64(x / a) - Float64(t * Float64(y / Float64(a * a)))) / z), -1.0, Float64(y / a));
	else
		tmp = fma(x, Float64(fma(a, Float64(z / t), 1.0) / t), Float64(Float64(y * z) / Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.5e-29], N[Not[LessEqual[a, 4100000.0]], $MachinePrecision]], N[(N[(N[(N[(x / a), $MachinePrecision] - N[(t * N[(y / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -1.0 + N[(y / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(a * N[(z / t), $MachinePrecision] + 1.0), $MachinePrecision] / t), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5000000000000001e-29 or 4.1e6 < a

   1. Initial program 72.8%

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

   3. Taylor expanded in z around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -1.5000000000000001e-29 < a < 4.1e6

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 63.1

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. pow2 N/A

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

       3. inv-pow N/A

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

       4. lower-fma.f64 N/A

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

       5. inv-pow N/A

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

       6. associate-/r* N/A

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

       7. div-add N/A

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

       8. lower-/.f64 N/A

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

       9. +-commutative N/A

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

       10. associate-/l* N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. mul-1-neg N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-/.f64 N/A

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

   11. Applied rewrites 76.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-29} \lor \neg \left(a \leq 4100000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{a} - t \cdot \frac{y}{a \cdot a}}{z}, -1, \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(a, \frac{z}{t}, 1\right)}{t}, \frac{y \cdot z}{-t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-31} \lor \neg \left(a \leq 4100000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a \cdot z}, -1, \frac{y}{a}\right) + \frac{t}{a \cdot a} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(a, \frac{z}{t}, 1\right)}{t}, \frac{y \cdot z}{-t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.2e-31) (not (<= a 4100000.0)))
   (+ (fma (/ x (* a z)) -1.0 (/ y a)) (* (/ t (* a a)) (/ y z)))
   (fma x (/ (fma a (/ z t) 1.0) t) (/ (* y z) (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.2e-31) || !(a <= 4100000.0)) {
		tmp = fma((x / (a * z)), -1.0, (y / a)) + ((t / (a * a)) * (y / z));
	} else {
		tmp = fma(x, (fma(a, (z / t), 1.0) / t), ((y * z) / -t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.2e-31) || !(a <= 4100000.0))
		tmp = Float64(fma(Float64(x / Float64(a * z)), -1.0, Float64(y / a)) + Float64(Float64(t / Float64(a * a)) * Float64(y / z)));
	else
		tmp = fma(x, Float64(fma(a, Float64(z / t), 1.0) / t), Float64(Float64(y * z) / Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.2e-31], N[Not[LessEqual[a, 4100000.0]], $MachinePrecision]], N[(N[(N[(x / N[(a * z), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(y / a), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(a * N[(z / t), $MachinePrecision] + 1.0), $MachinePrecision] / t), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.20000000000000018e-31 or 4.1e6 < a

   1. Initial program 72.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if -3.20000000000000018e-31 < a < 4.1e6

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 63.1

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. pow2 N/A

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

       3. inv-pow N/A

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

       4. lower-fma.f64 N/A

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

       5. inv-pow N/A

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

       6. associate-/r* N/A

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

       7. div-add N/A

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

       8. lower-/.f64 N/A

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

       9. +-commutative N/A

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

       10. associate-/l* N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. mul-1-neg N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-/.f64 N/A

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

   11. Applied rewrites 76.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-31} \lor \neg \left(a \leq 4100000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a \cdot z}, -1, \frac{y}{a}\right) + \frac{t}{a \cdot a} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(a, \frac{z}{t}, 1\right)}{t}, \frac{y \cdot z}{-t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-202} \lor \neg \left(t \leq 1.4 \cdot 10^{-174}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(a, \frac{z}{t}, 1\right)}{t}, \frac{y \cdot z}{-t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{a}, \frac{x - z \cdot y}{z \cdot z}, \frac{x}{z}\right) - y}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.3e-202) (not (<= t 1.4e-174)))
   (fma x (/ (fma a (/ z t) 1.0) t) (/ (* y z) (- t)))
   (/ (- (fma (/ t a) (/ (- x (* z y)) (* z z)) (/ x z)) y) (- a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.3e-202) || !(t <= 1.4e-174)) {
		tmp = fma(x, (fma(a, (z / t), 1.0) / t), ((y * z) / -t));
	} else {
		tmp = (fma((t / a), ((x - (z * y)) / (z * z)), (x / z)) - y) / -a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.3e-202) || !(t <= 1.4e-174))
		tmp = fma(x, Float64(fma(a, Float64(z / t), 1.0) / t), Float64(Float64(y * z) / Float64(-t)));
	else
		tmp = Float64(Float64(fma(Float64(t / a), Float64(Float64(x - Float64(z * y)) / Float64(z * z)), Float64(x / z)) - y) / Float64(-a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.3e-202], N[Not[LessEqual[t, 1.4e-174]], $MachinePrecision]], N[(x * N[(N[(a * N[(z / t), $MachinePrecision] + 1.0), $MachinePrecision] / t), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t / a), $MachinePrecision] * N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2999999999999999e-202 or 1.39999999999999999e-174 < t

   1. Initial program 86.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 59.2

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. pow2 N/A

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

       3. inv-pow N/A

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

       4. lower-fma.f64 N/A

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

       5. inv-pow N/A

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

       6. associate-/r* N/A

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

       7. div-add N/A

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

       8. lower-/.f64 N/A

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

       9. +-commutative N/A

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

       10. associate-/l* N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. mul-1-neg N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-/.f64 N/A

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

   11. Applied rewrites 66.7%

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

   if -2.2999999999999999e-202 < t < 1.39999999999999999e-174

   1. Initial program 83.0%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 57.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-202} \lor \neg \left(t \leq 1.4 \cdot 10^{-174}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(a, \frac{z}{t}, 1\right)}{t}, \frac{y \cdot z}{-t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{a}, \frac{x - z \cdot y}{z \cdot z}, \frac{x}{z}\right) - y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 48.6

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

5. Applied rewrites 48.6%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 46.4

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

8. Applied rewrites 46.4%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. pow2 N/A

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

    3. inv-pow N/A

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

    4. lower-fma.f64 N/A

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

    5. inv-pow N/A

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

    6. associate-/r* N/A

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

    7. div-add N/A

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

    8. lower-/.f64 N/A

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

    9. +-commutative N/A

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

    10. associate-/l* N/A

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

    11. lower-fma.f64 N/A

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

    12. lower-/.f64 N/A

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

    13. mul-1-neg N/A

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

    14. lower-neg.f64 N/A

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

    15. lower-/.f64 N/A

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

11. Applied rewrites 53.7%

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

12. Final simplification 53.7%

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

Alternative 7: 42.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 48.6

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

5. Applied rewrites 48.6%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 46.4

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

8. Applied rewrites 46.4%

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

Reproduce

herbie shell --seed 2025057 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 4392440296622287/125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))))))

  (/ (- x (* y z)) (- t (* a z))))