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

Percentage Accurate: 85.6% → 99.1%

Time: 3.5s

Alternatives: 10

Speedup: 0.7×

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: 85.6% 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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - a \cdot z\\ t_3 := \mathsf{fma}\left(\frac{z}{a \cdot z - t}, y, \frac{x}{t\_2}\right)\\ t_4 := \frac{t\_1}{t\_2}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{-79}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{z}{y \cdot z - x}, a, \frac{t}{t\_1}\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z)))
        (t_2 (- t (* a z)))
        (t_3 (fma (/ z (- (* a z) t)) y (/ x t_2)))
        (t_4 (/ t_1 t_2)))
   (if (<= t_4 -5e-79)
     t_3
     (if (<= t_4 2e-48)
       (/ 1.0 (fma (/ z (- (* y z) x)) a (/ t t_1)))
       (if (<= t_4 INFINITY) t_3 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (a * z);
	double t_3 = fma((z / ((a * z) - t)), y, (x / t_2));
	double t_4 = t_1 / t_2;
	double tmp;
	if (t_4 <= -5e-79) {
		tmp = t_3;
	} else if (t_4 <= 2e-48) {
		tmp = 1.0 / fma((z / ((y * z) - x)), a, (t / t_1));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t - Float64(a * z))
	t_3 = fma(Float64(z / Float64(Float64(a * z) - t)), y, Float64(x / t_2))
	t_4 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_4 <= -5e-79)
		tmp = t_3;
	elseif (t_4 <= 2e-48)
		tmp = Float64(1.0 / fma(Float64(z / Float64(Float64(y * z) - x)), a, Float64(t / t_1)));
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * y + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, -5e-79], t$95$3, If[LessEqual[t$95$4, 2e-48], N[(1.0 / N[(N[(z / N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] * a + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99999999999999999e-79 or 1.9999999999999999e-48 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 89.1%

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

   if -4.99999999999999999e-79 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.9999999999999999e-48

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. remove-double-neg N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower--.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. remove-double-neg N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 85.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 85.2

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

   3. Applied rewrites 85.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-neg-frac N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lift--.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 85.8

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

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

   1. Initial program 85.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 34.9

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

   4. Applied rewrites 34.9%

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

Alternative 2: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1}\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a}, y, t\_2\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t\_1}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a \cdot z - t}, z, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\frac{x}{z} - y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (/ x t_1)))
   (if (<= z -2.7e+128)
     (fma (/ 1.0 a) y t_2)
     (if (<= z 2e-61)
       (/ (fma (- z) y x) t_1)
       (if (<= z 4.4e+165)
         (fma (/ y (- (* a z) t)) z t_2)
         (* -1.0 (/ (- (/ x z) 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;
	double tmp;
	if (z <= -2.7e+128) {
		tmp = fma((1.0 / a), y, t_2);
	} else if (z <= 2e-61) {
		tmp = fma(-z, y, x) / t_1;
	} else if (z <= 4.4e+165) {
		tmp = fma((y / ((a * z) - t)), z, t_2);
	} else {
		tmp = -1.0 * (((x / z) - y) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(x / t_1)
	tmp = 0.0
	if (z <= -2.7e+128)
		tmp = fma(Float64(1.0 / a), y, t_2);
	elseif (z <= 2e-61)
		tmp = Float64(fma(Float64(-z), y, x) / t_1);
	elseif (z <= 4.4e+165)
		tmp = fma(Float64(y / Float64(Float64(a * z) - t)), z, t_2);
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(x / z) - 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[(x / t$95$1), $MachinePrecision]}, If[LessEqual[z, -2.7e+128], N[(N[(1.0 / a), $MachinePrecision] * y + t$95$2), $MachinePrecision], If[LessEqual[z, 2e-61], N[(N[((-z) * y + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 4.4e+165], N[(N[(y / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * z + t$95$2), $MachinePrecision], N[(-1.0 * N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.70000000000000001e128

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 89.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 65.1

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

   6. Applied rewrites 65.1%

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

   if -2.70000000000000001e128 < z < 2.0000000000000001e-61

   1. Initial program 85.6%

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

      1. *-lft-identity N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

   3. Applied rewrites 85.6%

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

   if 2.0000000000000001e-61 < z < 4.3999999999999998e165

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. mult-flip N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 86.4%

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

   if 4.3999999999999998e165 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{z} - y}{a}} \]
   8. 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 50.5

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

   9. Applied rewrites 50.5%

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

Alternative 3: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - a \cdot z}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a}, y, t\_1\right)\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a \cdot z - t}, y, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\frac{x}{z} - y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* a z)))))
   (if (<= z -5.5e+129)
     (fma (/ 1.0 a) y t_1)
     (if (<= z 4.7e+165)
       (fma (/ z (- (* a z) t)) y t_1)
       (* -1.0 (/ (- (/ x z) y) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (a * z));
	double tmp;
	if (z <= -5.5e+129) {
		tmp = fma((1.0 / a), y, t_1);
	} else if (z <= 4.7e+165) {
		tmp = fma((z / ((a * z) - t)), y, t_1);
	} else {
		tmp = -1.0 * (((x / z) - y) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (z <= -5.5e+129)
		tmp = fma(Float64(1.0 / a), y, t_1);
	elseif (z <= 4.7e+165)
		tmp = fma(Float64(z / Float64(Float64(a * z) - t)), y, t_1);
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(x / z) - y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e+129], N[(N[(1.0 / a), $MachinePrecision] * y + t$95$1), $MachinePrecision], If[LessEqual[z, 4.7e+165], N[(N[(z / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * y + t$95$1), $MachinePrecision], N[(-1.0 * N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.49999999999999984e129

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 89.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 65.1

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

   6. Applied rewrites 65.1%

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

   if -5.49999999999999984e129 < z < 4.70000000000000016e165

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 89.1%

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

   if 4.70000000000000016e165 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{z} - y}{a}} \]
   8. 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 50.5

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

   9. Applied rewrites 50.5%

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

Alternative 4: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a}, y, \frac{x}{t - a \cdot z}\right)\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+165}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\frac{x}{z} - y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e+128)
   (fma (/ 1.0 a) y (/ x (- t (* a z))))
   (if (<= z 4.7e+165)
     (/ (- x (* y z)) (fma (- z) a t))
     (* -1.0 (/ (- (/ x z) y) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e+128) {
		tmp = fma((1.0 / a), y, (x / (t - (a * z))));
	} else if (z <= 4.7e+165) {
		tmp = (x - (y * z)) / fma(-z, a, t);
	} else {
		tmp = -1.0 * (((x / z) - y) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e+128)
		tmp = fma(Float64(1.0 / a), y, Float64(x / Float64(t - Float64(a * z))));
	elseif (z <= 4.7e+165)
		tmp = Float64(Float64(x - Float64(y * z)) / fma(Float64(-z), a, t));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(x / z) - y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e+128], N[(N[(1.0 / a), $MachinePrecision] * y + N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.7e+165], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.70000000000000001e128

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 89.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 65.1

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

   6. Applied rewrites 65.1%

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

   if -2.70000000000000001e128 < z < 4.70000000000000016e165

   1. Initial program 85.6%

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

      1. remove-double-neg N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. sub-flip N/A

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

      5. distribute-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 85.6

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

   3. Applied rewrites 85.6%

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

   if 4.70000000000000016e165 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{z} - y}{a}} \]
   8. 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 50.5

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

   9. Applied rewrites 50.5%

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

Alternative 5: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{\frac{x}{z} - y}{a}\\ \mathbf{if}\;z \leq -1.16 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+165}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* -1.0 (/ (- (/ x z) y) a))))
   (if (<= z -1.16e+122)
     t_1
     (if (<= z 4.7e+165) (/ (- x (* y z)) (fma (- z) a 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 <= -1.16e+122) {
		tmp = t_1;
	} else if (z <= 4.7e+165) {
		tmp = (x - (y * z)) / fma(-z, a, 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 <= -1.16e+122)
		tmp = t_1;
	elseif (z <= 4.7e+165)
		tmp = Float64(Float64(x - Float64(y * z)) / fma(Float64(-z), a, 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, -1.16e+122], t$95$1, If[LessEqual[z, 4.7e+165], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.16e122 or 4.70000000000000016e165 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{z} - y}{a}} \]
   8. 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 50.5

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

   9. Applied rewrites 50.5%

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

   if -1.16e122 < z < 4.70000000000000016e165

   1. Initial program 85.6%

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

      1. remove-double-neg N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. sub-flip N/A

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

      5. distribute-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 85.6

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

   3. Applied rewrites 85.6%

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

Alternative 6: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{\frac{x}{z} - y}{a}\\ \mathbf{if}\;z \leq -1.16 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+165}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* -1.0 (/ (- (/ x z) y) a))))
   (if (<= z -1.16e+122)
     t_1
     (if (<= z 4.7e+165) (/ (- x (* y z)) (- t (* a z))) 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 <= -1.16e+122) {
		tmp = t_1;
	} else if (z <= 4.7e+165) {
		tmp = (x - (y * z)) / (t - (a * z));
	} else {
		tmp = t_1;
	}
	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 = (-1.0d0) * (((x / z) - y) / a)
    if (z <= (-1.16d+122)) then
        tmp = t_1
    else if (z <= 4.7d+165) then
        tmp = (x - (y * z)) / (t - (a * z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static 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 <= -1.16e+122) {
		tmp = t_1;
	} else if (z <= 4.7e+165) {
		tmp = (x - (y * z)) / (t - (a * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -1.0 * (((x / z) - y) / a)
	tmp = 0
	if z <= -1.16e+122:
		tmp = t_1
	elif z <= 4.7e+165:
		tmp = (x - (y * z)) / (t - (a * z))
	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 <= -1.16e+122)
		tmp = t_1;
	elseif (z <= 4.7e+165)
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = -1.0 * (((x / z) - y) / a);
	tmp = 0.0;
	if (z <= -1.16e+122)
		tmp = t_1;
	elseif (z <= 4.7e+165)
		tmp = (x - (y * z)) / (t - (a * z));
	else
		tmp = t_1;
	end
	tmp_2 = 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, -1.16e+122], t$95$1, If[LessEqual[z, 4.7e+165], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.16e122 or 4.70000000000000016e165 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{z} - y}{a}} \]
   8. 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 50.5

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

   9. Applied rewrites 50.5%

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

   if -1.16e122 < z < 4.70000000000000016e165

   1. Initial program 85.6%

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

Alternative 7: 72.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{\frac{x}{z} - y}{a}\\ \mathbf{if}\;z \leq -3 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1450000:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* -1.0 (/ (- (/ x z) y) a))))
   (if (<= z -3e-48) t_1 (if (<= z 1450000.0) (/ x (- t (* a z))) 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 <= -3e-48) {
		tmp = t_1;
	} else if (z <= 1450000.0) {
		tmp = x / (t - (a * z));
	} else {
		tmp = t_1;
	}
	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 = (-1.0d0) * (((x / z) - y) / a)
    if (z <= (-3d-48)) then
        tmp = t_1
    else if (z <= 1450000.0d0) then
        tmp = x / (t - (a * z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static 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 <= -3e-48) {
		tmp = t_1;
	} else if (z <= 1450000.0) {
		tmp = x / (t - (a * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -1.0 * (((x / z) - y) / a)
	tmp = 0
	if z <= -3e-48:
		tmp = t_1
	elif z <= 1450000.0:
		tmp = x / (t - (a * z))
	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 <= -3e-48)
		tmp = t_1;
	elseif (z <= 1450000.0)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = -1.0 * (((x / z) - y) / a);
	tmp = 0.0;
	if (z <= -3e-48)
		tmp = t_1;
	elseif (z <= 1450000.0)
		tmp = x / (t - (a * z));
	else
		tmp = t_1;
	end
	tmp_2 = 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, -3e-48], t$95$1, If[LessEqual[z, 1450000.0], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9999999999999999e-48 or 1.45e6 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{z} - y}{a}} \]
   8. 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 50.5

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

   9. Applied rewrites 50.5%

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

   if -2.9999999999999999e-48 < z < 1.45e6

   1. Initial program 85.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 54.7

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

   4. Applied rewrites 54.7%

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

Alternative 8: 65.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8200000:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.8e+125)
   (/ y a)
   (if (<= z 8200000.0) (/ x (- t (* a z))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.8e+125) {
		tmp = y / a;
	} else if (z <= 8200000.0) {
		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 <= (-6.8d+125)) then
        tmp = y / a
    else if (z <= 8200000.0d0) 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 <= -6.8e+125) {
		tmp = y / a;
	} else if (z <= 8200000.0) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.8e+125:
		tmp = y / a
	elif z <= 8200000.0:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.8e+125)
		tmp = Float64(y / a);
	elseif (z <= 8200000.0)
		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 <= -6.8e+125)
		tmp = y / a;
	elseif (z <= 8200000.0)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.8e+125], N[(y / a), $MachinePrecision], If[LessEqual[z, 8200000.0], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.7999999999999998e125 or 8.2e6 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 34.9

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

   4. Applied rewrites 34.9%

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

   if -6.7999999999999998e125 < z < 8.2e6

   1. Initial program 85.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 54.7

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

   4. Applied rewrites 54.7%

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

Alternative 9: 54.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.6e-52) (/ y a) (if (<= z 1.9e-54) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.6e-52) {
		tmp = y / a;
	} else if (z <= 1.9e-54) {
		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 <= (-8.6d-52)) then
        tmp = y / a
    else if (z <= 1.9d-54) 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 <= -8.6e-52) {
		tmp = y / a;
	} else if (z <= 1.9e-54) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.6e-52:
		tmp = y / a
	elif z <= 1.9e-54:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.6e-52)
		tmp = Float64(y / a);
	elseif (z <= 1.9e-54)
		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 <= -8.6e-52)
		tmp = y / a;
	elseif (z <= 1.9e-54)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.6e-52], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.9e-54], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.6000000000000007e-52 or 1.9000000000000001e-54 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 34.9

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

   4. Applied rewrites 34.9%

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

   if -8.6000000000000007e-52 < z < 1.9000000000000001e-54

   1. Initial program 85.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 36.5

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

   4. Applied rewrites 36.5%

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

Alternative 10: 36.5% accurate, 3.6× 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 85.6%

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

2. Taylor expanded in z around 0

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

   1. lower-/.f64 36.5

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

4. Applied rewrites 36.5%

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

Reproduce

herbie shell --seed 2025154 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64
  (/ (- x (* y z)) (- t (* a z))))