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

Percentage Accurate: 95.9% → 99.8%

Time: 3.5s

Alternatives: 8

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x (- (/ y z) t)) z)))
   (if (<= (* z t) (- INFINITY))
     t_1
     (if (<= (* z t) 1e+165) (/ x (fma (- z) t y)) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y / z) - t)) / z;
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((z * t) <= 1e+165) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(Float64(y / z) - t)) / z)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+165)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+165], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 9.99999999999999899e164 < (*.f64 z t)

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-/.f64 N/A

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

      13. inv-pow N/A

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

      14. pow-plus N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\frac{x \cdot {y}^{\color{blue}{0}}}{\mathsf{neg}\left(\left(-z\right)\right)}}{\mathsf{neg}\left(\left(t + \frac{y}{-z}\right)\right)} \]

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. *-lft-identity N/A

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

      19. lower-/.f64 N/A

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

      20. lift-neg.f64 N/A

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

      21. remove-double-neg N/A

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

      22. lift-+.f64 N/A

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

   7. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. mult-flip-rev N/A

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

      19. lower-/.f64 84.7

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

   9. Applied rewrites 84.7%

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

   if -inf.0 < (*.f64 z t) < 9.99999999999999899e164

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-1}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1}{z}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (* (/ x z) (/ -1.0 t))
   (if (<= (* z t) 5e+293) (/ x (fma (- z) t y)) (/ (* x (/ -1.0 z)) t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (x / z) * (-1.0 / t);
	} else if ((z * t) <= 5e+293) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = (x * (-1.0 / z)) / t;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(x / z) * Float64(-1.0 / t));
	elseif (Float64(z * t) <= 5e+293)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / z)) / t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[(x / z), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+293], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. mult-flip N/A

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

   10. Applied rewrites 52.1%

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

   if -inf.0 < (*.f64 z t) < 5.00000000000000033e293

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   if 5.00000000000000033e293 < (*.f64 z t)

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{x \cdot {y}^{\color{blue}{0}}}{z}\right)}{t} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. metadata-eval N/A

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

      18. distribute-frac-neg2 N/A

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

      19. metadata-eval N/A

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

      20. lift-neg.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-neg.f64 N/A

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

      24. frac-2neg-rev N/A

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

      25. lower-/.f64 52.1

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

   10. Applied rewrites 52.1%

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

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-1}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (* z t))))
   (if (<= t_1 (- INFINITY))
     (/ (/ (- x) z) t)
     (if (<= t_1 2e+307) (/ x (fma (- z) t y)) (* (/ x z) (/ -1.0 t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = y - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-x / z) / t;
	} else if (t_1 <= 2e+307) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = (x / z) * (-1.0 / t);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(y - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (t_1 <= 2e+307)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(x / z) * Float64(-1.0 / t));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 2e+307], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 y (*.f64 z t)) < -inf.0

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{x \cdot {y}^{\color{blue}{0}}}{z}\right)}{t} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. distribute-neg-frac N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 52.2

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

   10. Applied rewrites 52.2%

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

   if -inf.0 < (-.f64 y (*.f64 z t)) < 1.99999999999999997e307

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   if 1.99999999999999997e307 < (-.f64 y (*.f64 z t))

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. mult-flip N/A

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

   10. Applied rewrites 52.1%

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

Alternative 4: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := y - z \cdot t\\ t_2 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (* z t))) (t_2 (/ (/ (- x) z) t)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 2e+307) (/ x (fma (- z) t y)) t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = y - (z * t);
	double t_2 = (-x / z) / t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 2e+307) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(y - Float64(z * t))
	t_2 = Float64(Float64(Float64(-x) / z) / t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 2e+307)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 2e+307], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 y (*.f64 z t)) < -inf.0 or 1.99999999999999997e307 < (-.f64 y (*.f64 z t))

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{x \cdot {y}^{\color{blue}{0}}}{z}\right)}{t} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. distribute-neg-frac N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 52.2

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

   10. Applied rewrites 52.2%

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

   if -inf.0 < (-.f64 y (*.f64 z t)) < 1.99999999999999997e307

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

Alternative 5: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := y - z \cdot t\\ t_2 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (* z t))) (t_2 (/ (/ (- x) z) t)))
   (if (<= t_1 (- INFINITY)) t_2 (if (<= t_1 2e+307) (/ x t_1) t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = y - (z * t);
	double t_2 = (-x / z) / t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 2e+307) {
		tmp = x / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = y - (z * t);
	double t_2 = (-x / z) / t;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= 2e+307) {
		tmp = x / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = y - (z * t)
	t_2 = (-x / z) / t
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= 2e+307:
		tmp = x / t_1
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(y - Float64(z * t))
	t_2 = Float64(Float64(Float64(-x) / z) / t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 2e+307)
		tmp = Float64(x / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = y - (z * t);
	t_2 = (-x / z) / t;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= 2e+307)
		tmp = x / t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 2e+307], N[(x / t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 y (*.f64 z t)) < -inf.0 or 1.99999999999999997e307 < (-.f64 y (*.f64 z t))

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{x \cdot {y}^{\color{blue}{0}}}{z}\right)}{t} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. distribute-neg-frac N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 52.2

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

   10. Applied rewrites 52.2%

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

   if -inf.0 < (-.f64 y (*.f64 z t)) < 1.99999999999999997e307

   1. Initial program 95.9%

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

Alternative 6: 78.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (- x) z) t)))
   (if (<= (* z t) -1e-53) t_1 (if (<= (* z t) 2e+63) (/ x y) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -1e-53) {
		tmp = t_1;
	} else if ((z * t) <= 2e+63) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-x / z) / t
    if ((z * t) <= (-1d-53)) then
        tmp = t_1
    else if ((z * t) <= 2d+63) then
        tmp = x / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -1e-53) {
		tmp = t_1;
	} else if ((z * t) <= 2e+63) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (-x / z) / t
	tmp = 0
	if (z * t) <= -1e-53:
		tmp = t_1
	elif (z * t) <= 2e+63:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-x) / z) / t)
	tmp = 0.0
	if (Float64(z * t) <= -1e-53)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+63)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (-x / z) / t;
	tmp = 0.0;
	if ((z * t) <= -1e-53)
		tmp = t_1;
	elseif ((z * t) <= 2e+63)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e-53], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+63], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000003e-53 or 2.00000000000000012e63 < (*.f64 z t)

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{x \cdot {y}^{\color{blue}{0}}}{z}\right)}{t} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. distribute-neg-frac N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 52.2

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

   10. Applied rewrites 52.2%

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

   if -1.00000000000000003e-53 < (*.f64 z t) < 2.00000000000000012e63

   1. Initial program 95.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 53.9

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

   4. Applied rewrites 53.9%

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

Alternative 7: 62.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{t \cdot z}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* t z))))
   (if (<= (* z t) -5e+215) t_1 (if (<= (* z t) 2e+104) (/ x y) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / (t * z);
	double tmp;
	if ((z * t) <= -5e+215) {
		tmp = t_1;
	} else if ((z * t) <= 2e+104) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (t * z)
    if ((z * t) <= (-5d+215)) then
        tmp = t_1
    else if ((z * t) <= 2d+104) then
        tmp = x / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / (t * z);
	double tmp;
	if ((z * t) <= -5e+215) {
		tmp = t_1;
	} else if ((z * t) <= 2e+104) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / (t * z)
	tmp = 0
	if (z * t) <= -5e+215:
		tmp = t_1
	elif (z * t) <= 2e+104:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -5e+215)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+104)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / (t * z);
	tmp = 0.0;
	if ((z * t) <= -5e+215)
		tmp = t_1;
	elseif ((z * t) <= 2e+104)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+215], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+104], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000001e215 or 2e104 < (*.f64 z t)

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z \cdot t\right)\right) + \color{blue}{\left|e^{\log y}\right|}} \]

      6. rem-exp-log N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. rem-exp-log N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(z\right)\right) \cdot t + \left|\color{blue}{e^{\log y}}\right|} \]

      10. exp-fabs N/A

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

      11. rem-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 21.0

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

   8. Applied rewrites 21.0%

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

   if -5.0000000000000001e215 < (*.f64 z t) < 2e104

   1. Initial program 95.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 53.9

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

   4. Applied rewrites 53.9%

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

Alternative 8: 53.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x y))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return x / y;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return x / y

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x / y)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / y;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

2. Taylor expanded in y around inf

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

   1. lower-/.f64 53.9

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

4. Applied rewrites 53.9%

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

Reproduce

herbie shell --seed 2025150 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64
  (/ x (- y (* z t))))