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

Percentage Accurate: 85.9% → 94.5%

Time: 4.2s

Alternatives: 11

Speedup: 0.8×

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: 85.9% 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: 94.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1} - y \cdot \frac{z}{t\_1}\\ t_3 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{-307}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{a} - t \cdot \frac{y}{a \cdot a}}{z}, -1, \frac{y}{a}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

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 -4e-307)
     t_2
     (if (<= t_3 0.0)
       (fma (/ (- (/ x a) (* t (/ y (* a a)))) z) -1.0 (/ y a))
       (if (<= t_3 INFINITY) t_2 (/ y a))))))

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 <= -4e-307) {
		tmp = t_2;
	} 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 = y / a;
	}
	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 <= -4e-307)
		tmp = t_2;
	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(y / a);
	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, -4e-307], t$95$2, 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[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -3.99999999999999964e-307 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 94.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. 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 97.5

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

   3. Applied rewrites 97.5%

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

   if -3.99999999999999964e-307 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 61.9%

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

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z} + \frac{y}{a}} \]
   3. 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 74.9

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

   4. Applied rewrites 74.9%

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

   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. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ \mathbf{if}\;z \leq 1.25 \cdot 10^{+203}:\\ \;\;\;\;\frac{x}{t\_1} - y \cdot \frac{z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(1 + -1 \cdot \frac{x}{y \cdot z}\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))))
   (if (<= z 1.25e+203)
     (- (/ x t_1) (* y (/ z t_1)))
     (/ (* y (+ 1.0 (* -1.0 (/ x (* y z))))) a))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double tmp;
	if (z <= 1.25e+203) {
		tmp = (x / t_1) - (y * (z / t_1));
	} else {
		tmp = (y * (1.0 + (-1.0 * (x / (y * z))))) / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t - (a * z)
    if (z <= 1.25d+203) then
        tmp = (x / t_1) - (y * (z / t_1))
    else
        tmp = (y * (1.0d0 + ((-1.0d0) * (x / (y * z))))) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double tmp;
	if (z <= 1.25e+203) {
		tmp = (x / t_1) - (y * (z / t_1));
	} else {
		tmp = (y * (1.0 + (-1.0 * (x / (y * z))))) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	tmp = 0
	if z <= 1.25e+203:
		tmp = (x / t_1) - (y * (z / t_1))
	else:
		tmp = (y * (1.0 + (-1.0 * (x / (y * z))))) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	tmp = 0.0
	if (z <= 1.25e+203)
		tmp = Float64(Float64(x / t_1) - Float64(y * Float64(z / t_1)));
	else
		tmp = Float64(Float64(y * Float64(1.0 + Float64(-1.0 * Float64(x / Float64(y * z))))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	tmp = 0.0;
	if (z <= 1.25e+203)
		tmp = (x / t_1) - (y * (z / t_1));
	else
		tmp = (y * (1.0 + (-1.0 * (x / (y * z))))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.25e+203], N[(N[(x / t$95$1), $MachinePrecision] - N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(1.0 + N[(-1.0 * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.24999999999999999e203

   1. Initial program 88.8%

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

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

   3. Applied rewrites 90.8%

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

   if 1.24999999999999999e203 < z

   1. Initial program 55.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

Alternative 3: 68.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-z}{t - a \cdot z} \cdot y\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z) (- t (* a z))) y)))
   (if (<= z -7.8e-47)
     t_1
     (if (<= z 3e+18)
       (/ x (fma (- a) z t))
       (if (<= z 1.25e+203) t_1 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-z / (t - (a * z))) * y;
	double tmp;
	if (z <= -7.8e-47) {
		tmp = t_1;
	} else if (z <= 3e+18) {
		tmp = x / fma(-a, z, t);
	} else if (z <= 1.25e+203) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-z) / Float64(t - Float64(a * z))) * y)
	tmp = 0.0
	if (z <= -7.8e-47)
		tmp = t_1;
	elseif (z <= 3e+18)
		tmp = Float64(x / fma(Float64(-a), z, t));
	elseif (z <= 1.25e+203)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-z) / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -7.8e-47], t$95$1, If[LessEqual[z, 3e+18], N[(x / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e+203], t$95$1, N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.79999999999999956e-47 or 3e18 < z < 1.24999999999999999e203

   1. Initial program 76.5%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 79.6

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 30.8

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

   7. Applied rewrites 30.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \frac{z}{t - a \cdot z}\right) \cdot y \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 59.9

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

   10. Applied rewrites 59.9%

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

   if -7.79999999999999956e-47 < z < 3e18

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 75.1%

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

   if 1.24999999999999999e203 < z

   1. Initial program 55.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 71.8

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

   4. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{\frac{x}{z} - y}{a}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* -1.0 (/ (- (/ x z) y) a))))
   (if (<= z -5.5e+75)
     t_1
     (if (<= z 8.6e+141) (/ (- x (* y z)) (fma (- a) z t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -1.0 * (((x / z) - y) / a);
	double tmp;
	if (z <= -5.5e+75) {
		tmp = t_1;
	} else if (z <= 8.6e+141) {
		tmp = (x - (y * z)) / fma(-a, z, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(-1.0 * Float64(Float64(Float64(x / z) - y) / a))
	tmp = 0.0
	if (z <= -5.5e+75)
		tmp = t_1;
	elseif (z <= 8.6e+141)
		tmp = Float64(Float64(x - Float64(y * z)) / fma(Float64(-a), z, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(-1.0 * N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e+75], t$95$1, If[LessEqual[z, 8.6e+141], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5000000000000001e75 or 8.5999999999999997e141 < z

   1. Initial program 62.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. 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 74.9

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

   3. Applied rewrites 74.9%

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

   4. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 79.3

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

   6. Applied rewrites 79.3%

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

   if -5.5000000000000001e75 < z < 8.5999999999999997e141

   1. Initial program 97.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 97.0

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

   3. Applied rewrites 97.0%

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

Alternative 5: 66.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+75}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-47}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 4.05 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.06e+75)
   (/ y a)
   (if (<= z -6e-47)
     (/ (- x (* y z)) t)
     (if (<= z 4.05e+34) (/ x (fma (- a) z t)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.06e+75) {
		tmp = y / a;
	} else if (z <= -6e-47) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 4.05e+34) {
		tmp = x / fma(-a, z, t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.06e+75)
		tmp = Float64(y / a);
	elseif (z <= -6e-47)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 4.05e+34)
		tmp = Float64(x / fma(Float64(-a), z, t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.06e+75], N[(y / a), $MachinePrecision], If[LessEqual[z, -6e-47], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 4.05e+34], N[(x / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.06e75 or 4.05e34 < z

   1. Initial program 66.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 60.7

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

   4. Applied rewrites 60.7%

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

   if -1.06e75 < z < -6.00000000000000033e-47

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 47.6%

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

   if -6.00000000000000033e-47 < z < 4.05e34

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 74.6%

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

Alternative 6: 87.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+203}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 1.45e+203) (/ (- x (* y z)) (fma (- a) z t)) (/ y a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 1.45e+203) {
		tmp = (x - (y * z)) / fma(-a, z, t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 1.45e+203)
		tmp = Float64(Float64(x - Float64(y * z)) / fma(Float64(-a), z, t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 1.45e+203], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.45000000000000005e203

   1. Initial program 88.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 88.8

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

   3. Applied rewrites 88.8%

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

   if 1.45000000000000005e203 < z

   1. Initial program 55.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 71.8

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

   4. Applied rewrites 71.8%

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

Alternative 7: 87.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+203}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 1.45e+203) (/ (- x (* y z)) (- t (* a z))) (/ y a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 1.45e+203) {
		tmp = (x - (y * z)) / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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
    real(8) :: tmp
    if (z <= 1.45d+203) then
        tmp = (x - (y * z)) / (t - (a * z))
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 1.45e+203) {
		tmp = (x - (y * z)) / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= 1.45e+203:
		tmp = (x - (y * z)) / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 1.45e+203)
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 1.45e+203)
		tmp = (x - (y * z)) / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 1.45e+203], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.45000000000000005e203

   1. Initial program 88.8%

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

   if 1.45000000000000005e203 < z

   1. Initial program 55.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 71.8

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

   4. Applied rewrites 71.8%

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

Alternative 8: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{+101}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.05 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.38e+101)
   (/ y a)
   (if (<= z 4.05e+34) (/ x (fma (- a) z t)) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.38e+101) {
		tmp = y / a;
	} else if (z <= 4.05e+34) {
		tmp = x / fma(-a, z, t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.38e+101)
		tmp = Float64(y / a);
	elseif (z <= 4.05e+34)
		tmp = Float64(x / fma(Float64(-a), z, t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.38e+101], N[(y / a), $MachinePrecision], If[LessEqual[z, 4.05e+34], N[(x / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.38e101 or 4.05e34 < z

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 61.8

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

   4. Applied rewrites 61.8%

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

   if -1.38e101 < z < 4.05e34

   1. Initial program 98.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 98.3

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 69.1%

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

Alternative 9: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{+101}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.05 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.38e+101)
   (/ y a)
   (if (<= z 4.05e+34) (/ x (- t (* a z))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.38e+101) {
		tmp = y / a;
	} else if (z <= 4.05e+34) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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
    real(8) :: tmp
    if (z <= (-1.38d+101)) then
        tmp = y / a
    else if (z <= 4.05d+34) then
        tmp = x / (t - (a * z))
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.38e+101) {
		tmp = y / a;
	} else if (z <= 4.05e+34) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.38e+101:
		tmp = y / a
	elif z <= 4.05e+34:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.38e+101)
		tmp = Float64(y / a);
	elseif (z <= 4.05e+34)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.38e+101)
		tmp = y / a;
	elseif (z <= 4.05e+34)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.38e+101], N[(y / a), $MachinePrecision], If[LessEqual[z, 4.05e+34], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.38e101 or 4.05e34 < z

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 61.8

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

   4. Applied rewrites 61.8%

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

   if -1.38e101 < z < 4.05e34

   1. Initial program 98.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 69.1%

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

Alternative 10: 55.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+60) (/ y a) (if (<= z 6.5e-24) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+60) {
		tmp = y / a;
	} else if (z <= 6.5e-24) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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
    real(8) :: tmp
    if (z <= (-2.3d+60)) then
        tmp = y / a
    else if (z <= 6.5d-24) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+60) {
		tmp = y / a;
	} else if (z <= 6.5e-24) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e+60:
		tmp = y / a
	elif z <= 6.5e-24:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+60)
		tmp = Float64(y / a);
	elseif (z <= 6.5e-24)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e+60)
		tmp = y / a;
	elseif (z <= 6.5e-24)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+60], N[(y / a), $MachinePrecision], If[LessEqual[z, 6.5e-24], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.30000000000000017e60 or 6.5e-24 < z

   1. Initial program 70.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 56.9

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

   4. Applied rewrites 56.9%

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

   if -2.30000000000000017e60 < z < 6.5e-24

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 53.6

         \[\leadsto \frac{x}{\color{blue}{t}} \]

   4. Applied rewrites 53.6%

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

Alternative 11: 35.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{t} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

2. Taylor expanded in z around 0

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

   1. lower-/.f64 35.6

      \[\leadsto \frac{x}{\color{blue}{t}} \]

4. Applied rewrites 35.6%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
5. Add Preprocessing

Developer Target 1: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (- (/ x t_1) (/ y (- (/ t z) a)))))
   (if (< z -32113435955957344.0)
     t_2
     (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 t_1)) t_2))))

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 / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static 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 / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x / t_1) - (y / ((t / z) - a))
	tmp = 0
	if z < -32113435955957344.0:
		tmp = t_2
	elif z < 3.5139522372978296e-86:
		tmp = (x - (y * z)) * (1.0 / t_1)
	else:
		tmp = t_2
	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(Float64(t / z) - a)))
	tmp = 0.0
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = Float64(Float64(x - Float64(y * z)) * Float64(1.0 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = 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[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2025106 
(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))))