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

Percentage Accurate: 74.8% → 88.1%

Time: 19.5s

Alternatives: 7

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / 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, b)
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), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

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

Initial Program: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / 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, b)
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), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

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

Alternative 1: 88.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\frac{\frac{y}{t}}{t\_2} + \frac{\frac{x}{z}}{t\_2}\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (fma b (/ y t) (+ 1.0 a))))
   (if (<= t_1 (- INFINITY))
     (* (+ (/ (/ y t) t_2) (/ (/ x z) t_2)) z)
     (if (<= t_1 1e+301) t_1 (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(b, (y / t), (1.0 + a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((y / t) / t_2) + ((x / z) / t_2)) * z;
	} else if (t_1 <= 1e+301) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = fma(b, Float64(y / t), Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(y / t) / t_2) + Float64(Float64(x / z) / t_2)) * z);
	elseif (t_1 <= 1e+301)
		tmp = t_1;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(y / t), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(x / z), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 1e+301], t$95$1, N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 38.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000005e301

   1. Initial program 91.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if 1.00000000000000005e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 86.6

         \[\leadsto \frac{z}{\color{blue}{b}} \]

   5. Applied rewrites 86.6%

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

Alternative 2: 74.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{\frac{y}{t}}{t\_2} + \frac{\frac{x}{z}}{t\_2}\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot \frac{z}{-1 - a} + \frac{\left(y \cdot x\right) \cdot b}{{\left(1 + a\right)}^{2}}}{t}, -1, \frac{x}{1 + a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (fma b (/ y t) (+ 1.0 a))))
   (if (<= t_1 2e+28)
     (* (+ (/ (/ y t) t_2) (/ (/ x z) t_2)) z)
     (if (<= t_1 INFINITY)
       (fma
        (/ (+ (* y (/ z (- -1.0 a))) (/ (* (* y x) b) (pow (+ 1.0 a) 2.0))) t)
        -1.0
        (/ x (+ 1.0 a)))
       (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(b, (y / t), (1.0 + a));
	double tmp;
	if (t_1 <= 2e+28) {
		tmp = (((y / t) / t_2) + ((x / z) / t_2)) * z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((((y * (z / (-1.0 - a))) + (((y * x) * b) / pow((1.0 + a), 2.0))) / t), -1.0, (x / (1.0 + a)));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = fma(b, Float64(y / t), Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= 2e+28)
		tmp = Float64(Float64(Float64(Float64(y / t) / t_2) + Float64(Float64(x / z) / t_2)) * z);
	elseif (t_1 <= Inf)
		tmp = fma(Float64(Float64(Float64(y * Float64(z / Float64(-1.0 - a))) + Float64(Float64(Float64(y * x) * b) / (Float64(1.0 + a) ^ 2.0))) / t), -1.0, Float64(x / Float64(1.0 + a)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+28], N[(N[(N[(N[(y / t), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(x / z), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(y * N[(z / N[(-1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y * x), $MachinePrecision] * b), $MachinePrecision] / N[Power[N[(1.0 + a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * -1.0 + N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999992e28

   1. Initial program 85.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   if 1.99999999999999992e28 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 87.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 72.1%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto \frac{z}{\color{blue}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{\frac{y}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)} + \frac{\frac{x}{z}}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\right) \cdot z\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot \frac{z}{-1 - a} + \frac{\left(y \cdot x\right) \cdot b}{{\left(1 + a\right)}^{2}}}{t}, -1, \frac{x}{1 + a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{\frac{y}{t}}{t\_2} + \frac{\frac{x}{z}}{t\_2}\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{y \cdot z}{-1 - a} + \frac{b \cdot \left(x \cdot y\right)}{{\left(1 + a\right)}^{2}}}{t}, \frac{x}{1 + a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (fma b (/ y t) (+ 1.0 a))))
   (if (<= t_1 2e+28)
     (* (+ (/ (/ y t) t_2) (/ (/ x z) t_2)) z)
     (if (<= t_1 1e+301)
       (fma
        -1.0
        (/ (+ (/ (* y z) (- -1.0 a)) (/ (* b (* x y)) (pow (+ 1.0 a) 2.0))) t)
        (/ x (+ 1.0 a)))
       (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(b, (y / t), (1.0 + a));
	double tmp;
	if (t_1 <= 2e+28) {
		tmp = (((y / t) / t_2) + ((x / z) / t_2)) * z;
	} else if (t_1 <= 1e+301) {
		tmp = fma(-1.0, ((((y * z) / (-1.0 - a)) + ((b * (x * y)) / pow((1.0 + a), 2.0))) / t), (x / (1.0 + a)));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = fma(b, Float64(y / t), Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= 2e+28)
		tmp = Float64(Float64(Float64(Float64(y / t) / t_2) + Float64(Float64(x / z) / t_2)) * z);
	elseif (t_1 <= 1e+301)
		tmp = fma(-1.0, Float64(Float64(Float64(Float64(y * z) / Float64(-1.0 - a)) + Float64(Float64(b * Float64(x * y)) / (Float64(1.0 + a) ^ 2.0))) / t), Float64(x / Float64(1.0 + a)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+28], N[(N[(N[(N[(y / t), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(x / z), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 1e+301], N[(-1.0 * N[(N[(N[(N[(y * z), $MachinePrecision] / N[(-1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[Power[N[(1.0 + a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999992e28

   1. Initial program 85.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   if 1.99999999999999992e28 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000005e301

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in t around -inf

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

      1. lower-fma.f64 N/A

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

   8. Applied rewrites 76.2%

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

   if 1.00000000000000005e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 86.6

         \[\leadsto \frac{z}{\color{blue}{b}} \]

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{\frac{y}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)} + \frac{\frac{x}{z}}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\right) \cdot z\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{y \cdot z}{-1 - a} + \frac{b \cdot \left(x \cdot y\right)}{{\left(1 + a\right)}^{2}}}{t}, \frac{x}{1 + a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+82}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+158}:\\ \;\;\;\;\left(\frac{\frac{y}{t}}{t\_1} + \frac{\frac{x}{z}}{t\_1}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b (/ y t) (+ 1.0 a))))
   (if (<= y -1.85e+82)
     (/ (+ z (/ (* t x) y)) b)
     (if (<= y 5.2e+158) (* (+ (/ (/ y t) t_1) (/ (/ x z) t_1)) z) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, (y / t), (1.0 + a));
	double tmp;
	if (y <= -1.85e+82) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (y <= 5.2e+158) {
		tmp = (((y / t) / t_1) + ((x / z) / t_1)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, Float64(y / t), Float64(1.0 + a))
	tmp = 0.0
	if (y <= -1.85e+82)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (y <= 5.2e+158)
		tmp = Float64(Float64(Float64(Float64(y / t) / t_1) + Float64(Float64(x / z) / t_1)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.85e+82], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 5.2e+158], N[(N[(N[(N[(y / t), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(x / z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8500000000000001e82

   1. Initial program 50.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 58.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      4. lift-*.f64 75.7

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

   8. Applied rewrites 75.7%

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

   if -1.8500000000000001e82 < y < 5.2e158

   1. Initial program 92.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   if 5.2e158 < y

   1. Initial program 43.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 68.5

         \[\leadsto \frac{z}{\color{blue}{b}} \]

   5. Applied rewrites 68.5%

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

Alternative 5: 72.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+82} \lor \neg \left(y \leq 4.3 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{y}{t}}{t\_1} + \frac{\frac{x}{z}}{t\_1}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b (/ y t) (+ 1.0 a))))
   (if (or (<= y -1.85e+82) (not (<= y 4.3e+158)))
     (/ (+ z (/ (* t x) y)) b)
     (* (+ (/ (/ y t) t_1) (/ (/ x z) t_1)) z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, (y / t), (1.0 + a));
	double tmp;
	if ((y <= -1.85e+82) || !(y <= 4.3e+158)) {
		tmp = (z + ((t * x) / y)) / b;
	} else {
		tmp = (((y / t) / t_1) + ((x / z) / t_1)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, Float64(y / t), Float64(1.0 + a))
	tmp = 0.0
	if ((y <= -1.85e+82) || !(y <= 4.3e+158))
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	else
		tmp = Float64(Float64(Float64(Float64(y / t) / t_1) + Float64(Float64(x / z) / t_1)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -1.85e+82], N[Not[LessEqual[y, 4.3e+158]], $MachinePrecision]], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[(N[(y / t), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(x / z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8500000000000001e82 or 4.3e158 < y

   1. Initial program 47.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      4. lift-*.f64 70.6

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

   8. Applied rewrites 70.6%

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

   if -1.8500000000000001e82 < y < 4.3e158

   1. Initial program 92.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+82} \lor \neg \left(y \leq 4.3 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{y}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)} + \frac{\frac{x}{z}}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

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, b)
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), intent (in) :: b
    code = (z + ((t * x) / y)) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
2. Add Preprocessing

3. Taylor expanded in y around -inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 32.0%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

   4. lift-*.f64 42.1

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

8. Applied rewrites 42.1%

   \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
9. Add Preprocessing

Alternative 7: 37.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{z \cdot \left(1 + \frac{t \cdot x}{y \cdot z}\right)}{b} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* z (+ 1.0 (/ (* t x) (* y z)))) b))

C

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

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, b)
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), intent (in) :: b
    code = (z * (1.0d0 + ((t * x) / (y * z)))) / b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (z * (1.0 + ((t * x) / (y * z)))) / b;
}

Python

def code(x, y, z, t, a, b):
	return (z * (1.0 + ((t * x) / (y * z)))) / b

Julia

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (z * (1.0 + ((t * x) / (y * z)))) / b;
end

Wolfram

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

Derivation

1. Initial program 77.5%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 66.2%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 38.9

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

8. Applied rewrites 38.9%

   \[\leadsto \frac{z \cdot \left(1 + \frac{t \cdot x}{y \cdot z}\right)}{\color{blue}{b}} \]
9. Add Preprocessing

Reproduce

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

  :alt
  (! :herbie-platform default (if (< t -1707385670788761/12500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 1518483551868623/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))