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

Percentage Accurate: 85.6% → 92.6%

Time: 7.5s

Alternatives: 11

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: 92.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{t\_1} \cdot y\\ \mathbf{elif}\;t\_2 \leq 10^{+291}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 (- INFINITY))
     (* (/ (- (/ x y) z) t_1) y)
     (if (<= t_2 1e+291) t_2 (/ (- (/ 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 - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (((x / y) - z) / t_1) * y;
	} else if (t_2 <= 1e+291) {
		tmp = t_2;
	} else {
		tmp = ((x / z) - y) / -a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (((x / y) - z) / t_1) * y;
	} else if (t_2 <= 1e+291) {
		tmp = t_2;
	} else {
		tmp = ((x / z) - y) / -a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x - (y * z)) / t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (((x / y) - z) / t_1) * y
	elif t_2 <= 1e+291:
		tmp = t_2
	else:
		tmp = ((x / z) - y) / -a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(x / y) - z) / t_1) * y);
	elseif (t_2 <= 1e+291)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(x / z) - y) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (((x / y) - z) / t_1) * y;
	elseif (t_2 <= 1e+291)
		tmp = t_2;
	else
		tmp = ((x / z) - y) / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[(x / y), $MachinePrecision] - z), $MachinePrecision] / t$95$1), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, 1e+291], t$95$2, N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.9999999999999996e290

   1. Initial program 94.2%

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

   if 9.9999999999999996e290 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 26.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 49.6

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in a around inf

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

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

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 96.3

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

   7. Applied rewrites 96.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{t - a \cdot z} \cdot y\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 10^{+291}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{x}{z} - y}{-a}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(a, z, -t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (/ x z) y) (- a))))
   (if (<= z -1.45e+51)
     t_1
     (if (<= z 2.2e-71)
       (/ x (- t (* a z)))
       (if (<= z 5.4e+50) (* (/ y (fma a z (- t))) z) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x / z) - y) / -a;
	double tmp;
	if (z <= -1.45e+51) {
		tmp = t_1;
	} else if (z <= 2.2e-71) {
		tmp = x / (t - (a * z));
	} else if (z <= 5.4e+50) {
		tmp = (y / fma(a, z, -t)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x / z) - y) / Float64(-a))
	tmp = 0.0
	if (z <= -1.45e+51)
		tmp = t_1;
	elseif (z <= 2.2e-71)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	elseif (z <= 5.4e+50)
		tmp = Float64(Float64(y / fma(a, z, Float64(-t))) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / (-a)), $MachinePrecision]}, If[LessEqual[z, -1.45e+51], t$95$1, If[LessEqual[z, 2.2e-71], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.4e+50], N[(N[(y / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4499999999999999e51 or 5.4e50 < z

   1. Initial program 61.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 71.1

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

   4. Applied rewrites 71.1%

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

   5. Taylor expanded in a around inf

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

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

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if -1.4499999999999999e51 < z < 2.19999999999999997e-71

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 2.19999999999999997e-71 < z < 5.4e50

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-neg.f64 76.6

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 76.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(a, z, -t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+59}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.8e+91)
   (/ y a)
   (if (<= z -1.3e+46)
     (/ (- x) (* z a))
     (if (<= z 1.8e-78)
       (/ x t)
       (if (<= z 1.85e+59) (* (- y) (/ z t)) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.8e+91) {
		tmp = y / a;
	} else if (z <= -1.3e+46) {
		tmp = -x / (z * a);
	} else if (z <= 1.8e-78) {
		tmp = x / t;
	} else if (z <= 1.85e+59) {
		tmp = -y * (z / 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 <= (-8.8d+91)) then
        tmp = y / a
    else if (z <= (-1.3d+46)) then
        tmp = -x / (z * a)
    else if (z <= 1.8d-78) then
        tmp = x / t
    else if (z <= 1.85d+59) then
        tmp = -y * (z / 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 <= -8.8e+91) {
		tmp = y / a;
	} else if (z <= -1.3e+46) {
		tmp = -x / (z * a);
	} else if (z <= 1.8e-78) {
		tmp = x / t;
	} else if (z <= 1.85e+59) {
		tmp = -y * (z / t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.8e+91:
		tmp = y / a
	elif z <= -1.3e+46:
		tmp = -x / (z * a)
	elif z <= 1.8e-78:
		tmp = x / t
	elif z <= 1.85e+59:
		tmp = -y * (z / t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.8e+91)
		tmp = Float64(y / a);
	elseif (z <= -1.3e+46)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 1.8e-78)
		tmp = Float64(x / t);
	elseif (z <= 1.85e+59)
		tmp = Float64(Float64(-y) * Float64(z / 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 <= -8.8e+91)
		tmp = y / a;
	elseif (z <= -1.3e+46)
		tmp = -x / (z * a);
	elseif (z <= 1.8e-78)
		tmp = x / t;
	elseif (z <= 1.85e+59)
		tmp = -y * (z / t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.8e+91], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.3e+46], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e-78], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.85e+59], N[((-y) * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.79999999999999998e91 or 1.84999999999999999e59 < z

   1. Initial program 57.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -8.79999999999999998e91 < z < -1.30000000000000007e46

   1. Initial program 92.1%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-neg.f64 84.3

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.7%

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

   if -1.30000000000000007e46 < z < 1.8000000000000001e-78

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if 1.8000000000000001e-78 < z < 1.84999999999999999e59

   1. Initial program 96.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+59}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(a, z, -t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e+92)
   (/ y a)
   (if (<= z 2.2e-71)
     (/ x (- t (* a z)))
     (if (<= z 4.5e+183) (* (/ y (fma a z (- t))) z) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e+92) {
		tmp = y / a;
	} else if (z <= 2.2e-71) {
		tmp = x / (t - (a * z));
	} else if (z <= 4.5e+183) {
		tmp = (y / fma(a, z, -t)) * z;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e+92)
		tmp = Float64(y / a);
	elseif (z <= 2.2e-71)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	elseif (z <= 4.5e+183)
		tmp = Float64(Float64(y / fma(a, z, Float64(-t))) * z);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e+92], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.2e-71], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e+183], N[(N[(y / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e92 or 4.50000000000000017e183 < z

   1. Initial program 52.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if -1.35e92 < z < 2.19999999999999997e-71

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if 2.19999999999999997e-71 < z < 4.50000000000000017e183

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-neg.f64 64.5

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

   5. Applied rewrites 64.5%

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

   7. Applied rewrites 69.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(a, z, -t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+183}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, z, -t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e+92)
   (/ y a)
   (if (<= z 3.7e-62)
     (/ x (- t (* a z)))
     (if (<= z 4.5e+183) (* y (/ z (fma a z (- t)))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e+92) {
		tmp = y / a;
	} else if (z <= 3.7e-62) {
		tmp = x / (t - (a * z));
	} else if (z <= 4.5e+183) {
		tmp = y * (z / fma(a, z, -t));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e+92)
		tmp = Float64(y / a);
	elseif (z <= 3.7e-62)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	elseif (z <= 4.5e+183)
		tmp = Float64(y * Float64(z / fma(a, z, Float64(-t))));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e+92], N[(y / a), $MachinePrecision], If[LessEqual[z, 3.7e-62], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e+183], N[(y * N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e92 or 4.50000000000000017e183 < z

   1. Initial program 52.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if -1.35e92 < z < 3.6999999999999998e-62

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if 3.6999999999999998e-62 < z < 4.50000000000000017e183

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-neg.f64 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 69.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+183}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, z, -t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.3 \cdot 10^{+170} \lor \neg \left(z \leq 4.5 \cdot 10^{+183}\right):\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7.3e+170) (not (<= z 4.5e+183)))
   (/ (- (/ x z) y) (- a))
   (/ (- x (* y z)) (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7.3e+170) || !(z <= 4.5e+183)) {
		tmp = ((x / z) - y) / -a;
	} else {
		tmp = (x - (y * z)) / (t - (a * z));
	}
	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 <= (-7.3d+170)) .or. (.not. (z <= 4.5d+183))) then
        tmp = ((x / z) - y) / -a
    else
        tmp = (x - (y * z)) / (t - (a * z))
    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 <= -7.3e+170) || !(z <= 4.5e+183)) {
		tmp = ((x / z) - y) / -a;
	} else {
		tmp = (x - (y * z)) / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7.3e+170) or not (z <= 4.5e+183):
		tmp = ((x / z) - y) / -a
	else:
		tmp = (x - (y * z)) / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7.3e+170) || !(z <= 4.5e+183))
		tmp = Float64(Float64(Float64(x / z) - y) / Float64(-a));
	else
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -7.3e+170) || ~((z <= 4.5e+183)))
		tmp = ((x / z) - y) / -a;
	else
		tmp = (x - (y * z)) / (t - (a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7.3e+170], N[Not[LessEqual[z, 4.5e+183]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.30000000000000037e170 or 4.50000000000000017e183 < z

   1. Initial program 47.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 60.6

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

   4. Applied rewrites 60.6%

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

   5. Taylor expanded in a around inf

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

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

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 87.3

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

   7. Applied rewrites 87.3%

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

   if -7.30000000000000037e170 < z < 4.50000000000000017e183

   1. Initial program 95.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.3 \cdot 10^{+170} \lor \neg \left(z \leq 4.5 \cdot 10^{+183}\right):\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-246}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e+92)
   (/ y a)
   (if (<= z 1.2e-246)
     (/ x (- t (* a z)))
     (if (<= z 5.7e+51) (/ (- x (* z y)) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e+92) {
		tmp = y / a;
	} else if (z <= 1.2e-246) {
		tmp = x / (t - (a * z));
	} else if (z <= 5.7e+51) {
		tmp = (x - (z * y)) / 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 <= (-1.35d+92)) then
        tmp = y / a
    else if (z <= 1.2d-246) then
        tmp = x / (t - (a * z))
    else if (z <= 5.7d+51) then
        tmp = (x - (z * y)) / 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 <= -1.35e+92) {
		tmp = y / a;
	} else if (z <= 1.2e-246) {
		tmp = x / (t - (a * z));
	} else if (z <= 5.7e+51) {
		tmp = (x - (z * y)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.35e+92:
		tmp = y / a
	elif z <= 1.2e-246:
		tmp = x / (t - (a * z))
	elif z <= 5.7e+51:
		tmp = (x - (z * y)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e+92)
		tmp = Float64(y / a);
	elseif (z <= 1.2e-246)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	elseif (z <= 5.7e+51)
		tmp = Float64(Float64(x - Float64(z * y)) / 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 <= -1.35e+92)
		tmp = y / a;
	elseif (z <= 1.2e-246)
		tmp = x / (t - (a * z));
	elseif (z <= 5.7e+51)
		tmp = (x - (z * y)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e+92], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.2e-246], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.7e+51], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e92 or 5.7000000000000002e51 < z

   1. Initial program 57.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.35e92 < z < 1.1999999999999999e-246

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if 1.1999999999999999e-246 < z < 5.7000000000000002e51

   1. Initial program 99.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-246}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+46}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+59}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.85e+46)
   (/ y a)
   (if (<= z 1.8e-78) (/ x t) (if (<= z 1.85e+59) (* (- y) (/ z t)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e+46) {
		tmp = y / a;
	} else if (z <= 1.8e-78) {
		tmp = x / t;
	} else if (z <= 1.85e+59) {
		tmp = -y * (z / 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 <= (-1.85d+46)) then
        tmp = y / a
    else if (z <= 1.8d-78) then
        tmp = x / t
    else if (z <= 1.85d+59) then
        tmp = -y * (z / 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 <= -1.85e+46) {
		tmp = y / a;
	} else if (z <= 1.8e-78) {
		tmp = x / t;
	} else if (z <= 1.85e+59) {
		tmp = -y * (z / t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.85e+46:
		tmp = y / a
	elif z <= 1.8e-78:
		tmp = x / t
	elif z <= 1.85e+59:
		tmp = -y * (z / t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.85e+46)
		tmp = Float64(y / a);
	elseif (z <= 1.8e-78)
		tmp = Float64(x / t);
	elseif (z <= 1.85e+59)
		tmp = Float64(Float64(-y) * Float64(z / 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 <= -1.85e+46)
		tmp = y / a;
	elseif (z <= 1.8e-78)
		tmp = x / t;
	elseif (z <= 1.85e+59)
		tmp = -y * (z / t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.85e+46], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.8e-78], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.85e+59], N[((-y) * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.84999999999999995e46 or 1.84999999999999999e59 < z

   1. Initial program 61.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if -1.84999999999999995e46 < z < 1.8000000000000001e-78

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if 1.8000000000000001e-78 < z < 1.84999999999999999e59

   1. Initial program 96.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.4%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+46}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+59}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+92} \lor \neg \left(z \leq 9 \cdot 10^{+134}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.35e+92) (not (<= z 9e+134))) (/ y a) (/ x (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.35e+92) || !(z <= 9e+134)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (a * z));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-1.35d+92)) .or. (.not. (z <= 9d+134))) then
        tmp = y / a
    else
        tmp = x / (t - (a * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.35e+92) || !(z <= 9e+134)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.35e+92) or not (z <= 9e+134):
		tmp = y / a
	else:
		tmp = x / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.35e+92) || !(z <= 9e+134))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(a * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.35e+92) || ~((z <= 9e+134)))
		tmp = y / a;
	else
		tmp = x / (t - (a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.35e+92], N[Not[LessEqual[z, 9e+134]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e92 or 8.9999999999999995e134 < z

   1. Initial program 54.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if -1.35e92 < z < 8.9999999999999995e134

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+92} \lor \neg \left(z \leq 9 \cdot 10^{+134}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+46} \lor \neg \left(z \leq 1.8 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.85e+46) (not (<= z 1.8e+33))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.85e+46) || !(z <= 1.8e+33)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-1.85d+46)) .or. (.not. (z <= 1.8d+33))) then
        tmp = y / a
    else
        tmp = x / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.85e+46) || !(z <= 1.8e+33)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.85e+46) or not (z <= 1.8e+33):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.85e+46) || !(z <= 1.8e+33))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.85e+46) || ~((z <= 1.8e+33)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.85e+46], N[Not[LessEqual[z, 1.8e+33]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.84999999999999995e46 or 1.8000000000000001e33 < z

   1. Initial program 63.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if -1.84999999999999995e46 < z < 1.8000000000000001e33

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 59.2

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

   5. Applied rewrites 59.2%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+46} \lor \neg \left(z \leq 1.8 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 36.8

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

5. Applied rewrites 36.8%

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

6. Final simplification 36.8%

   \[\leadsto \frac{x}{t} \]
7. Add Preprocessing

Developer Target 1: 97.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024350 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

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

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