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

Percentage Accurate: 74.8% → 89.7%

Time: 10.4s

Alternatives: 14

Speedup: 0.3×

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: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(1 + a, t, b \cdot y\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{y}{t} + \frac{x}{z}}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (* y (/ z (fma (+ 1.0 a) t (* b y))))
     (if (<= t_1 2e+287)
       t_1
       (if (<= t_1 INFINITY)
         (* (/ (+ (/ y t) (/ x z)) (fma (/ y t) b (+ 1.0 a))) z)
         (/ 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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (z / fma((1.0 + a), t, (b * y)));
	} else if (t_1 <= 2e+287) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((y / t) + (x / z)) / fma((y / t), b, (1.0 + a))) * z;
	} 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)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / fma(Float64(1.0 + a), t, Float64(b * y))));
	elseif (t_1 <= 2e+287)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(y / t) + Float64(x / z)) / fma(Float64(y / t), b, Float64(1.0 + a))) * z);
	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]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(z / N[(N[(1.0 + a), $MachinePrecision] * t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+287], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(y / t), $MachinePrecision] + N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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 31.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 21.9

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

   8. Applied rewrites 21.9%

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

   10. Applied rewrites 90.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 90.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000002e287

   1. Initial program 95.3%

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

   if 2.0000000000000002e287 < (/.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 32.1%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

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

   1. 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

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

   5. Applied rewrites 100.0%

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

Alternative 2: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(1 + a, t, b \cdot y\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (* y (/ z (fma (+ 1.0 a) t (* b y))))
     (if (<= t_1 1e+292) 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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (z / fma((1.0 + a), t, (b * y)));
	} else if (t_1 <= 1e+292) {
		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)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / fma(Float64(1.0 + a), t, Float64(b * y))));
	elseif (t_1 <= 1e+292)
		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]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(z / N[(N[(1.0 + a), $MachinePrecision] * t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+292], 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 31.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 21.9

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

   8. Applied rewrites 21.9%

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

   10. Applied rewrites 90.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 90.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1e292

   1. Initial program 95.3%

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

   if 1e292 < (/.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 75.2

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

   5. Applied rewrites 75.2%

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

Alternative 3: 87.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (* y (/ z (fma (+ 1.0 a) t (* b y))))
     (if (<= t_1 INFINITY)
       (/ (fma (/ y t) z x) (fma (/ y t) b (+ 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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (z / fma((1.0 + a), t, (b * y)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((y / t), z, x) / fma((y / t), b, (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)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / fma(Float64(1.0 + a), t, Float64(b * y))));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(y / t), z, x) / fma(Float64(y / t), b, 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(z / N[(N[(1.0 + a), $MachinePrecision] * t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(N[(y / t), $MachinePrecision] * b + 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))) < -inf.0

   1. Initial program 31.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 21.9

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

   8. Applied rewrites 21.9%

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

   10. Applied rewrites 90.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 90.6%

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

   if -inf.0 < (/.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 88.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 87.5

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

   5. Applied rewrites 87.5%

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

   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

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

   5. Applied rewrites 100.0%

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

Alternative 4: 84.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (* y (/ z (fma (+ 1.0 a) t (* b y))))
     (if (<= t_1 INFINITY)
       (/ (fma (/ y t) z x) (fma (/ b t) y (+ 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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (z / fma((1.0 + a), t, (b * y)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((y / t), z, x) / fma((b / t), y, (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)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / fma(Float64(1.0 + a), t, Float64(b * y))));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(y / t), z, x) / fma(Float64(b / t), y, 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(z / N[(N[(1.0 + a), $MachinePrecision] * t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + 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))) < -inf.0

   1. Initial program 31.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 21.9

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

   8. Applied rewrites 21.9%

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

   10. Applied rewrites 90.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 90.6%

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

   if -inf.0 < (/.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 88.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 87.5

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

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 83.7%

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

   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

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

   5. Applied rewrites 100.0%

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

Alternative 5: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-110} \lor \neg \left(t \leq 2.1 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(1 + a, t, b \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -8e+106)
   (/ (fma (/ y t) z x) (+ 1.0 a))
   (if (or (<= t -2.25e-110) (not (<= t 2.1e-23)))
     (/ x (fma (/ y t) b (+ 1.0 a)))
     (* y (/ z (fma (+ 1.0 a) t (* b y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -8e+106) {
		tmp = fma((y / t), z, x) / (1.0 + a);
	} else if ((t <= -2.25e-110) || !(t <= 2.1e-23)) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else {
		tmp = y * (z / fma((1.0 + a), t, (b * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -8e+106)
		tmp = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a));
	elseif ((t <= -2.25e-110) || !(t <= 2.1e-23))
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	else
		tmp = Float64(y * Float64(z / fma(Float64(1.0 + a), t, Float64(b * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -8e+106], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -2.25e-110], N[Not[LessEqual[t, 2.1e-23]], $MachinePrecision]], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(N[(1.0 + a), $MachinePrecision] * t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.00000000000000073e106

   1. Initial program 79.8%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 89.2

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

   5. Applied rewrites 89.2%

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

   if -8.00000000000000073e106 < t < -2.25e-110 or 2.1000000000000001e-23 < t

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 78.0

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

   5. Applied rewrites 78.0%

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

   if -2.25e-110 < t < 2.1000000000000001e-23

   1. Initial program 62.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 36.4

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

   8. Applied rewrites 36.4%

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

   10. Applied rewrites 49.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 67.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-110} \lor \neg \left(t \leq 2.1 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(1 + a, t, b \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-110} \lor \neg \left(t \leq 2.1 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(1 + a, t, b \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2.25e-110) (not (<= t 2.1e-23)))
   (/ x (fma (/ y t) b (+ 1.0 a)))
   (* y (/ z (fma (+ 1.0 a) t (* b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2.25e-110) || !(t <= 2.1e-23)) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else {
		tmp = y * (z / fma((1.0 + a), t, (b * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -2.25e-110) || !(t <= 2.1e-23))
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	else
		tmp = Float64(y * Float64(z / fma(Float64(1.0 + a), t, Float64(b * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2.25e-110], N[Not[LessEqual[t, 2.1e-23]], $MachinePrecision]], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(N[(1.0 + a), $MachinePrecision] * t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.25e-110 or 2.1000000000000001e-23 < t

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -2.25e-110 < t < 2.1000000000000001e-23

   1. Initial program 62.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 36.4

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

   8. Applied rewrites 36.4%

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

   10. Applied rewrites 49.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 67.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-110} \lor \neg \left(t \leq 2.1 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(1 + a, t, b \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-17} \lor \neg \left(t \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(1 + a, t, b \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.9e-17) (not (<= t 3.2e-7)))
   (/ x (+ 1.0 a))
   (* y (/ z (fma (+ 1.0 a) t (* b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.9e-17) || !(t <= 3.2e-7)) {
		tmp = x / (1.0 + a);
	} else {
		tmp = y * (z / fma((1.0 + a), t, (b * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.9e-17) || !(t <= 3.2e-7))
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(y * Float64(z / fma(Float64(1.0 + a), t, Float64(b * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.9e-17], N[Not[LessEqual[t, 3.2e-7]], $MachinePrecision]], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(N[(1.0 + a), $MachinePrecision] * t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.9000000000000001e-17 or 3.2000000000000001e-7 < t

   1. Initial program 87.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. lower-+.f64 69.2

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -1.9000000000000001e-17 < t < 3.2000000000000001e-7

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 58.5

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 36.8

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

   8. Applied rewrites 36.8%

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

   10. Applied rewrites 46.8%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 63.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-17} \lor \neg \left(t \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(1 + a, t, b \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -470 \lor \neg \left(t \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -470.0) (not (<= t 3.2e-7)))
   (/ x (+ 1.0 a))
   (/ (fma t (/ x y) z) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -470.0) || !(t <= 3.2e-7)) {
		tmp = x / (1.0 + a);
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -470.0) || !(t <= 3.2e-7))
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -470.0], N[Not[LessEqual[t, 3.2e-7]], $MachinePrecision]], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -470 or 3.2000000000000001e-7 < t

   1. Initial program 88.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. lower-+.f64 71.3

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -470 < t < 3.2000000000000001e-7

   1. Initial program 67.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. lower-+.f64 N/A

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 61.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -470 \lor \neg \left(t \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot x - x, a, x\right)\\ \mathbf{elif}\;t \leq -470:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+226}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -5e+252)
   (fma (- (* a x) x) a x)
   (if (<= t -470.0)
     (/ x a)
     (if (<= t 2.4e+37) (/ z b) (if (<= t 6.8e+226) (/ x a) (/ x 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -5e+252) {
		tmp = fma(((a * x) - x), a, x);
	} else if (t <= -470.0) {
		tmp = x / a;
	} else if (t <= 2.4e+37) {
		tmp = z / b;
	} else if (t <= 6.8e+226) {
		tmp = x / a;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -5e+252)
		tmp = fma(Float64(Float64(a * x) - x), a, x);
	elseif (t <= -470.0)
		tmp = Float64(x / a);
	elseif (t <= 2.4e+37)
		tmp = Float64(z / b);
	elseif (t <= 6.8e+226)
		tmp = Float64(x / a);
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -5e+252], N[(N[(N[(a * x), $MachinePrecision] - x), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[t, -470.0], N[(x / a), $MachinePrecision], If[LessEqual[t, 2.4e+37], N[(z / b), $MachinePrecision], If[LessEqual[t, 6.8e+226], N[(x / a), $MachinePrecision], N[(x / 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.9999999999999997e252

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. lower-+.f64 79.5

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{a \cdot \left(a \cdot x - x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.9%

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

   if -4.9999999999999997e252 < t < -470 or 2.4e37 < t < 6.79999999999999958e226

   1. Initial program 88.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.2%

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

   if -470 < t < 2.4e37

   1. Initial program 68.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 53.7

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

   5. Applied rewrites 53.7%

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

   if 6.79999999999999958e226 < t

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. lower-+.f64 89.6

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{x}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.8%

      \[\leadsto \frac{x}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 42.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \mathbf{elif}\;t \leq -470:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+226}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -5e+252)
   (fma (- x) a x)
   (if (<= t -470.0)
     (/ x a)
     (if (<= t 2.4e+37) (/ z b) (if (<= t 6.8e+226) (/ x a) (/ x 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -5e+252) {
		tmp = fma(-x, a, x);
	} else if (t <= -470.0) {
		tmp = x / a;
	} else if (t <= 2.4e+37) {
		tmp = z / b;
	} else if (t <= 6.8e+226) {
		tmp = x / a;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -5e+252)
		tmp = fma(Float64(-x), a, x);
	elseif (t <= -470.0)
		tmp = Float64(x / a);
	elseif (t <= 2.4e+37)
		tmp = Float64(z / b);
	elseif (t <= 6.8e+226)
		tmp = Float64(x / a);
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -5e+252], N[((-x) * a + x), $MachinePrecision], If[LessEqual[t, -470.0], N[(x / a), $MachinePrecision], If[LessEqual[t, 2.4e+37], N[(z / b), $MachinePrecision], If[LessEqual[t, 6.8e+226], N[(x / a), $MachinePrecision], N[(x / 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.9999999999999997e252

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. lower-+.f64 79.5

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 58.0%

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

   if -4.9999999999999997e252 < t < -470 or 2.4e37 < t < 6.79999999999999958e226

   1. Initial program 88.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.2%

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

   if -470 < t < 2.4e37

   1. Initial program 68.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 53.7

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

   5. Applied rewrites 53.7%

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

   if 6.79999999999999958e226 < t

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. lower-+.f64 89.6

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{x}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.8%

      \[\leadsto \frac{x}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-21} \lor \neg \left(t \leq 3.4 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -6e-21) (not (<= t 3.4e-23))) (/ x (+ 1.0 a)) (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -6e-21) || !(t <= 3.4e-23)) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	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, 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
    real(8) :: tmp
    if ((t <= (-6d-21)) .or. (.not. (t <= 3.4d-23))) then
        tmp = x / (1.0d0 + a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -6e-21) || !(t <= 3.4e-23)) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -6e-21) or not (t <= 3.4e-23):
		tmp = x / (1.0 + a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -6e-21) || !(t <= 3.4e-23))
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -6e-21) || ~((t <= 3.4e-23)))
		tmp = x / (1.0 + a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -6e-21], N[Not[LessEqual[t, 3.4e-23]], $MachinePrecision]], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.99999999999999982e-21 or 3.4000000000000001e-23 < t

   1. Initial program 87.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. lower-+.f64 68.6

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   5. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -5.99999999999999982e-21 < t < 3.4000000000000001e-23

   1. Initial program 66.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-21} \lor \neg \left(t \leq 3.4 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -390000000 \lor \neg \left(a \leq 2.2 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -390000000.0) (not (<= a 2.2e-20))) (/ x a) (fma (- x) a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -390000000.0) || !(a <= 2.2e-20)) {
		tmp = x / a;
	} else {
		tmp = fma(-x, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -390000000.0) || !(a <= 2.2e-20))
		tmp = Float64(x / a);
	else
		tmp = fma(Float64(-x), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -390000000.0], N[Not[LessEqual[a, 2.2e-20]], $MachinePrecision]], N[(x / a), $MachinePrecision], N[((-x) * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.9e8 or 2.19999999999999991e-20 < a

   1. Initial program 73.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 60.9

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 46.1%

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

   if -3.9e8 < a < 2.19999999999999991e-20

   1. Initial program 80.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. lower-+.f64 43.0

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   5. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 42.9%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -390000000 \lor \neg \left(a \leq 2.2 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 19.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-x, a, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma (- x) a x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(-x, a, x);
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(-x), a, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[((-x) * a + x), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   2. lower-+.f64 44.8

      \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

5. Applied rewrites 44.8%

   \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

6. Taylor expanded in a around 0

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

8. Applied rewrites 22.9%

   \[\leadsto \mathsf{fma}\left(-x, \color{blue}{a}, x\right) \]
9. Add Preprocessing

Alternative 14: 4.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) \cdot a \end{array} \]

FPCore

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

C

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

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 * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[((-x) * a), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   2. lower-+.f64 44.8

      \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

5. Applied rewrites 44.8%

   \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

6. Taylor expanded in a around 0

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

8. Applied rewrites 22.9%

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

9. Taylor expanded in a around inf

   \[\leadsto -1 \cdot \left(a \cdot \color{blue}{x}\right) \]
10. Step-by-step derivation

11. Applied rewrites 4.7%

    \[\leadsto \left(-x\right) \cdot a \]
12. Add Preprocessing

Developer Target 1: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))
   (if (< t -1.3659085366310088e-271)
     t_1
     (if (< t 3.036967103737246e-130) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, 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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 * ((x + ((y / t) * z)) * (1.0d0 / ((a + 1.0d0) + ((y / t) * b))))
    if (t < (-1.3659085366310088d-271)) then
        tmp = t_1
    else if (t < 3.036967103737246d-130) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))))
	tmp = 0
	if t < -1.3659085366310088e-271:
		tmp = t_1
	elif t < 3.036967103737246e-130:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 * Float64(Float64(x + Float64(Float64(y / t) * z)) * Float64(1.0 / Float64(Float64(a + 1.0) + Float64(Float64(y / t) * b)))))
	tmp = 0.0
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	tmp = 0.0;
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 * N[(N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.3659085366310088e-271], t$95$1, If[Less[t, 3.036967103737246e-130], N[(z / b), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024359 
(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))))