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

Percentage Accurate: 74.8% → 86.9%

Time: 5.6s

Alternatives: 21

Speedup: 0.4×

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: 86.9% accurate, 0.2× speedup

Math

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

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 t) (/ z (+ 1.0 (+ a (/ (* b y) t)))))
     (if (<= t_1 -2e-165)
       t_1
       (if (<= t_1 INFINITY)
         (/ (fma y (/ z t) x) (fma b (/ y t) (+ 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 / t) * (z / (1.0 + (a + ((b * y) / t))));
	} else if (t_1 <= -2e-165) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), (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(Float64(y / t) * Float64(z / Float64(1.0 + Float64(a + Float64(Float64(b * y) / t)))));
	elseif (t_1 <= -2e-165)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), 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[(N[(y / t), $MachinePrecision] * N[(z / N[(1.0 + N[(a + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-165], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $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 30.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 57.2

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in x around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 78.5

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

   7. Applied rewrites 78.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2e-165

   1. Initial program 99.6%

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

   if -2e-165 < (/.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 80.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 79.4

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 80.8

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

   4. Applied rewrites 80.8%

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

   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 96.3

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

   5. Applied rewrites 96.3%

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

Alternative 2: 45.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+43}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+177}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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))
     (/ z b)
     (if (<= t_1 -1e+43)
       x
       (if (<= t_1 -5e-109)
         (/ x a)
         (if (<= t_1 0.0)
           (/ z b)
           (if (<= t_1 5e+177)
             (/ x a)
             (if (<= t_1 2e+301) (fma (- a) x x) (/ 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 = z / b;
	} else if (t_1 <= -1e+43) {
		tmp = x;
	} else if (t_1 <= -5e-109) {
		tmp = x / a;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 5e+177) {
		tmp = x / a;
	} else if (t_1 <= 2e+301) {
		tmp = fma(-a, x, x);
	} 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(z / b);
	elseif (t_1 <= -1e+43)
		tmp = x;
	elseif (t_1 <= -5e-109)
		tmp = Float64(x / a);
	elseif (t_1 <= 0.0)
		tmp = Float64(z / b);
	elseif (t_1 <= 5e+177)
		tmp = Float64(x / a);
	elseif (t_1 <= 2e+301)
		tmp = fma(Float64(-a), x, x);
	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[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -1e+43], x, If[LessEqual[t$95$1, -5e-109], N[(x / a), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 5e+177], N[(x / a), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], N[((-a) * x + x), $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 or -5.0000000000000002e-109 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 45.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000001e43

   1. Initial program 99.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 89.7

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 88.2

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift-+.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in a around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 39.3%

       \[\leadsto x \]

   if -1.00000000000000001e43 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.0000000000000002e-109 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5.0000000000000003e177

   1. Initial program 99.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 93.8

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 90.7

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift-+.f64 51.0

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

   7. Applied rewrites 51.0%

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

   8. Taylor expanded in a around inf

      \[\leadsto \frac{x}{a} \]
   9. Step-by-step derivation

   10. Applied rewrites 30.8%

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

   if 5.0000000000000003e177 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 87.9

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift-+.f64 70.4

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto -1 \cdot \left(a \cdot x\right) + x \]

      2. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot x + x \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot x + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), x, x\right) \]

      5. lower-neg.f64 54.4

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

   10. Applied rewrites 54.4%

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

Alternative 3: 56.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + a}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-101}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + t \cdot \frac{x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 (- INFINITY))
     (* (/ y t) (/ z (+ 1.0 a)))
     (if (<= t_2 -1e+43)
       t_1
       (if (<= t_2 -1e-101)
         (/ t_1 a)
         (if (<= t_2 0.0)
           (/ (+ z (/ (* t x) y)) b)
           (if (<= t_2 2e+301) (/ x (+ 1.0 a)) (/ (+ z (* t (/ x y))) b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / t) * (z / (1.0 + a));
	} else if (t_2 <= -1e+43) {
		tmp = t_1;
	} else if (t_2 <= -1e-101) {
		tmp = t_1 / a;
	} else if (t_2 <= 0.0) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (t_2 <= 2e+301) {
		tmp = x / (1.0 + a);
	} else {
		tmp = (z + (t * (x / y))) / b;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / t) * (z / (1.0 + a));
	} else if (t_2 <= -1e+43) {
		tmp = t_1;
	} else if (t_2 <= -1e-101) {
		tmp = t_1 / a;
	} else if (t_2 <= 0.0) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (t_2 <= 2e+301) {
		tmp = x / (1.0 + a);
	} else {
		tmp = (z + (t * (x / y))) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + ((y * z) / t)
	t_2 = t_1 / ((a + 1.0) + ((y * b) / t))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (y / t) * (z / (1.0 + a))
	elif t_2 <= -1e+43:
		tmp = t_1
	elif t_2 <= -1e-101:
		tmp = t_1 / a
	elif t_2 <= 0.0:
		tmp = (z + ((t * x) / y)) / b
	elif t_2 <= 2e+301:
		tmp = x / (1.0 + a)
	else:
		tmp = (z + (t * (x / y))) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) * Float64(z / Float64(1.0 + a)));
	elseif (t_2 <= -1e+43)
		tmp = t_1;
	elseif (t_2 <= -1e-101)
		tmp = Float64(t_1 / a);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (t_2 <= 2e+301)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z + Float64(t * Float64(x / y))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + ((y * z) / t);
	t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (y / t) * (z / (1.0 + a));
	elseif (t_2 <= -1e+43)
		tmp = t_1;
	elseif (t_2 <= -1e-101)
		tmp = t_1 / a;
	elseif (t_2 <= 0.0)
		tmp = (z + ((t * x) / y)) / b;
	elseif (t_2 <= 2e+301)
		tmp = x / (1.0 + a);
	else
		tmp = (z + (t * (x / y))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / t), $MachinePrecision] * N[(z / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e+43], t$95$1, If[LessEqual[t$95$2, -1e-101], N[(t$95$1 / a), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 2e+301], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(t * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 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 30.4%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in x around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-+.f64 50.4

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

   8. Applied rewrites 50.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000001e43

   1. Initial program 99.7%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if -1.00000000000000001e43 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000005e-101

   1. Initial program 99.5%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 43.5%

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

   if -1.00000000000000005e-101 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 70.2%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 51.7

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

   8. Applied rewrites 51.7%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 54.1

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

   5. Applied rewrites 54.1%

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

   if 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

       \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 4: 56.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + a}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + t \cdot \frac{x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 (- INFINITY))
     (* (/ y t) (/ z (+ 1.0 a)))
     (if (<= t_2 -1e+43)
       t_1
       (if (<= t_2 -1e-101)
         (/ (fma y (/ z t) x) a)
         (if (<= t_2 0.0)
           (/ (+ z (/ (* t x) y)) b)
           (if (<= t_2 2e+301) (/ x (+ 1.0 a)) (/ (+ z (* t (/ x y))) b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / t) * (z / (1.0 + a));
	} else if (t_2 <= -1e+43) {
		tmp = t_1;
	} else if (t_2 <= -1e-101) {
		tmp = fma(y, (z / t), x) / a;
	} else if (t_2 <= 0.0) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (t_2 <= 2e+301) {
		tmp = x / (1.0 + a);
	} else {
		tmp = (z + (t * (x / y))) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) * Float64(z / Float64(1.0 + a)));
	elseif (t_2 <= -1e+43)
		tmp = t_1;
	elseif (t_2 <= -1e-101)
		tmp = Float64(fma(y, Float64(z / t), x) / a);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (t_2 <= 2e+301)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z + Float64(t * Float64(x / y))) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / t), $MachinePrecision] * N[(z / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e+43], t$95$1, If[LessEqual[t$95$2, -1e-101], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 2e+301], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(t * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 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 30.4%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in x around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-+.f64 50.4

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

   8. Applied rewrites 50.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000001e43

   1. Initial program 99.7%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if -1.00000000000000001e43 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000005e-101

   1. Initial program 99.5%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 41.2

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

   5. Applied rewrites 41.2%

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

   if -1.00000000000000005e-101 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 70.2%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 51.7

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

   8. Applied rewrites 51.7%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 54.1

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

   5. Applied rewrites 54.1%

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

   if 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

       \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{+43}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-175}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z + t \cdot \frac{x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ 1.0 a)))
        (t_3 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_3 (- INFINITY))
     (* (/ y t) (/ z (fma (/ y t) b (+ a 1.0))))
     (if (<= t_3 -1e+43)
       t_2
       (if (<= t_3 5e-175)
         (/ (fma y (/ z t) x) (fma b (/ y t) a))
         (if (<= t_3 2e+301) t_2 (/ (+ z (* t (/ x y))) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / (1.0 + a);
	double t_3 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (y / t) * (z / fma((y / t), b, (a + 1.0)));
	} else if (t_3 <= -1e+43) {
		tmp = t_2;
	} else if (t_3 <= 5e-175) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), a);
	} else if (t_3 <= 2e+301) {
		tmp = t_2;
	} else {
		tmp = (z + (t * (x / y))) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(1.0 + a))
	t_3 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) * Float64(z / fma(Float64(y / t), b, Float64(a + 1.0))));
	elseif (t_3 <= -1e+43)
		tmp = t_2;
	elseif (t_3 <= 5e-175)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), a));
	elseif (t_3 <= 2e+301)
		tmp = t_2;
	else
		tmp = Float64(Float64(z + Float64(t * Float64(x / y))) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(y / t), $MachinePrecision] * N[(z / N[(N[(y / t), $MachinePrecision] * b + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e+43], t$95$2, If[LessEqual[t$95$3, 5e-175], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+301], t$95$2, N[(N[(z + N[(t * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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 30.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 57.2

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in x around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 66.4

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

   10. Applied rewrites 66.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000001e43 or 5e-175 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -1.00000000000000001e43 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5e-175

   1. Initial program 81.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 79.7

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 84.1

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

   4. Applied rewrites 84.1%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 67.9%

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

   if 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

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

Alternative 6: 54.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + t \cdot \frac{x}{y}}{b}\\ \end{array} \end{array} \]

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 -1e+43)
     (fma (/ z t) y x)
     (if (<= t_1 -1e-101)
       (/ (fma y (/ z t) x) a)
       (if (<= t_1 0.0)
         (/ (+ z (/ (* t x) y)) b)
         (if (<= t_1 2e+301) (/ x (+ 1.0 a)) (/ (+ z (* t (/ x y))) 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 <= -1e+43) {
		tmp = fma((z / t), y, x);
	} else if (t_1 <= -1e-101) {
		tmp = fma(y, (z / t), x) / a;
	} else if (t_1 <= 0.0) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (t_1 <= 2e+301) {
		tmp = x / (1.0 + a);
	} else {
		tmp = (z + (t * (x / y))) / 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 <= -1e+43)
		tmp = fma(Float64(z / t), y, x);
	elseif (t_1 <= -1e-101)
		tmp = Float64(fma(y, Float64(z / t), x) / a);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (t_1 <= 2e+301)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z + Float64(t * Float64(x / y))) / 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, -1e+43], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, -1e-101], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(t * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000001e43

   1. Initial program 77.7%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 47.1

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

   8. Applied rewrites 47.1%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-/.f64 40.6

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

   11. Applied rewrites 40.6%

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

   if -1.00000000000000001e43 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000005e-101

   1. Initial program 99.5%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 41.2

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

   5. Applied rewrites 41.2%

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

   if -1.00000000000000005e-101 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 70.2%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 51.7

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

   8. Applied rewrites 51.7%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 54.1

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

   5. Applied rewrites 54.1%

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

   if 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

       \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 -1e+43)
     (fma (/ z t) y x)
     (if (<= t_1 -5e-109)
       (/ (fma y (/ z t) x) a)
       (if (<= t_1 0.0)
         (/ z b)
         (if (<= t_1 2e+301) (/ x (+ 1.0 a)) (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -1e+43) {
		tmp = fma((z / t), y, x);
	} else if (t_1 <= -5e-109) {
		tmp = fma(y, (z / t), x) / a;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 2e+301) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= -1e+43)
		tmp = fma(Float64(z / t), y, x);
	elseif (t_1 <= -5e-109)
		tmp = Float64(fma(y, Float64(z / t), x) / a);
	elseif (t_1 <= 0.0)
		tmp = Float64(z / b);
	elseif (t_1 <= 2e+301)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+43], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, -5e-109], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], N[(x / N[(1.0 + a), $MachinePrecision]), $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))) < -1.00000000000000001e43

   1. Initial program 77.7%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 47.1

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

   8. Applied rewrites 47.1%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-/.f64 40.6

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

   11. Applied rewrites 40.6%

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

   if -1.00000000000000001e43 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.0000000000000002e-109

   1. Initial program 99.5%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 41.1

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

   5. Applied rewrites 41.1%

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

   if -5.0000000000000002e-109 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 47.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 54.1

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

   5. Applied rewrites 54.1%

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

Alternative 8: 54.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{x}{1 + a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-165}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (/ x (+ 1.0 a))))
   (if (<= t_1 -1e+54)
     (fma (/ z t) y x)
     (if (<= t_1 -2e-165)
       t_2
       (if (<= t_1 0.0) (/ z b) (if (<= t_1 2e+301) t_2 (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = x / (1.0 + a);
	double tmp;
	if (t_1 <= -1e+54) {
		tmp = fma((z / t), y, x);
	} else if (t_1 <= -2e-165) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 2e+301) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= -1e+54)
		tmp = fma(Float64(z / t), y, x);
	elseif (t_1 <= -2e-165)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(z / b);
	elseif (t_1 <= 2e+301)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+54], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, -2e-165], t$95$2, If[LessEqual[t$95$1, 0.0], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], t$95$2, N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.0000000000000001e54

   1. Initial program 76.6%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 46.4

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

   8. Applied rewrites 46.4%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-/.f64 41.1

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

   11. Applied rewrites 41.1%

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

   if -1.0000000000000001e54 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2e-165 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 53.2

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

   5. Applied rewrites 53.2%

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

   if -2e-165 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 43.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 61.3

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

   5. Applied rewrites 61.3%

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

Alternative 9: 57.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{x}{1 + a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-165}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (/ x (+ 1.0 a))))
   (if (<= t_1 (- INFINITY))
     (/ z b)
     (if (<= t_1 -2e-165)
       t_2
       (if (<= t_1 0.0) (/ z b) (if (<= t_1 2e+301) t_2 (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = x / (1.0 + a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_1 <= -2e-165) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 2e+301) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = x / (1.0 + a);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = z / b;
	} else if (t_1 <= -2e-165) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 2e+301) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	t_2 = x / (1.0 + a)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = z / b
	elif t_1 <= -2e-165:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = z / b
	elif t_1 <= 2e+301:
		tmp = t_2
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_1 <= -2e-165)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(z / b);
	elseif (t_1 <= 2e+301)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	t_2 = x / (1.0 + a);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = z / b;
	elseif (t_1 <= -2e-165)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = z / b;
	elseif (t_1 <= 2e+301)
		tmp = t_2;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -2e-165], t$95$2, If[LessEqual[t$95$1, 0.0], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], t$95$2, N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 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 or -2e-165 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 42.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 60.8

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

   5. Applied rewrites 60.8%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2e-165 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 54.6

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

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 41.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-175}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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))
     (/ z b)
     (if (<= t_1 -1e+35)
       x
       (if (<= t_1 5e-175) (/ z b) (if (<= t_1 2e+301) x (/ 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 = z / b;
	} else if (t_1 <= -1e+35) {
		tmp = x;
	} else if (t_1 <= 5e-175) {
		tmp = z / b;
	} else if (t_1 <= 2e+301) {
		tmp = x;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Java

public static 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.POSITIVE_INFINITY) {
		tmp = z / b;
	} else if (t_1 <= -1e+35) {
		tmp = x;
	} else if (t_1 <= 5e-175) {
		tmp = z / b;
	} else if (t_1 <= 2e+301) {
		tmp = x;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = z / b
	elif t_1 <= -1e+35:
		tmp = x
	elif t_1 <= 5e-175:
		tmp = z / b
	elif t_1 <= 2e+301:
		tmp = x
	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(z / b);
	elseif (t_1 <= -1e+35)
		tmp = x;
	elseif (t_1 <= 5e-175)
		tmp = Float64(z / b);
	elseif (t_1 <= 2e+301)
		tmp = x;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = z / b;
	elseif (t_1 <= -1e+35)
		tmp = x;
	elseif (t_1 <= 5e-175)
		tmp = z / b;
	elseif (t_1 <= 2e+301)
		tmp = x;
	else
		tmp = z / b;
	end
	tmp_2 = 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[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -1e+35], x, If[LessEqual[t$95$1, 5e-175], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], x, N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 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 or -9.9999999999999997e34 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5e-175 or 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 59.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 47.5

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

   5. Applied rewrites 47.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.9999999999999997e34 or 5e-175 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 91.3

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift-+.f64 55.6

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

   7. Applied rewrites 55.6%

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

   8. Taylor expanded in a around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 32.2%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{t \cdot x}{b}}{y}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + t \cdot \frac{x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 -5e-298)
     (/ (fma y (/ z t) x) (+ 1.0 a))
     (if (<= t_2 0.0)
       (+ (/ z b) (/ (/ (* t x) b) y))
       (if (<= t_2 2e+301) (/ t_1 (+ 1.0 a)) (/ (+ z (* t (/ x y))) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -5e-298) {
		tmp = fma(y, (z / t), x) / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = (z / b) + (((t * x) / b) / y);
	} else if (t_2 <= 2e+301) {
		tmp = t_1 / (1.0 + a);
	} else {
		tmp = (z + (t * (x / y))) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= -5e-298)
		tmp = Float64(fma(y, Float64(z / t), x) / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(t * x) / b) / y));
	elseif (t_2 <= 2e+301)
		tmp = Float64(t_1 / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z + Float64(t * Float64(x / y))) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-298], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(t * x), $MachinePrecision] / b), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+301], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(t * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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))) < -5.0000000000000002e-298

   1. Initial program 89.0%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -5.0000000000000002e-298 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 54.5%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in x around inf

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

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

Alternative 12: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + t \cdot \frac{x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 -5e-298)
     (/ (fma y (/ z t) x) (+ 1.0 a))
     (if (<= t_2 0.0)
       (/ (fma t (/ x y) z) b)
       (if (<= t_2 2e+301) (/ t_1 (+ 1.0 a)) (/ (+ z (* t (/ x y))) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -5e-298) {
		tmp = fma(y, (z / t), x) / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_2 <= 2e+301) {
		tmp = t_1 / (1.0 + a);
	} else {
		tmp = (z + (t * (x / y))) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= -5e-298)
		tmp = Float64(fma(y, Float64(z / t), x) / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_2 <= 2e+301)
		tmp = Float64(t_1 / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z + Float64(t * Float64(x / y))) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-298], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 2e+301], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(t * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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))) < -5.0000000000000002e-298

   1. Initial program 89.0%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -5.0000000000000002e-298 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 54.5%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

      1. +-commutative 68.0

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

      2. associate-*r/ 68.0

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

      3. +-commutative 68.0

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

      4. *-commutative 68.0

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

      5. associate-*r/ 68.0

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

      6. +-commutative 68.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 70.5

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

   10. Applied rewrites 70.5%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

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

Alternative 13: 72.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-298}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z + t \cdot \frac{x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (/ (fma y (/ z t) x) (+ 1.0 a))))
   (if (<= t_1 -5e-298)
     t_2
     (if (<= t_1 0.0)
       (/ (fma t (/ x y) z) b)
       (if (<= t_1 2e+301) t_2 (/ (+ z (* t (/ x y))) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(y, (z / t), x) / (1.0 + a);
	double tmp;
	if (t_1 <= -5e-298) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_1 <= 2e+301) {
		tmp = t_2;
	} else {
		tmp = (z + (t * (x / y))) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(fma(y, Float64(z / t), x) / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= -5e-298)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_1 <= 2e+301)
		tmp = t_2;
	else
		tmp = Float64(Float64(z + Float64(t * Float64(x / y))) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-298], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], t$95$2, N[(N[(z + N[(t * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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))) < -5.0000000000000002e-298 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 93.9%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 69.8

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

   5. Applied rewrites 69.8%

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

   if -5.0000000000000002e-298 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 54.5%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

      1. +-commutative 68.0

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

      2. associate-*r/ 68.0

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

      3. +-commutative 68.0

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

      4. *-commutative 68.0

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

      5. associate-*r/ 68.0

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

      6. +-commutative 68.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 70.5

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

   10. Applied rewrites 70.5%

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

   if 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

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

Alternative 14: 84.5% accurate, 0.3× speedup

Math

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

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 t) (/ z (+ 1.0 (+ a (/ (* b y) t)))))
     (if (<= t_1 INFINITY)
       (/ (fma y (/ z t) x) (fma b (/ y t) (+ 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 / t) * (z / (1.0 + (a + ((b * y) / t))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), (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(Float64(y / t) * Float64(z / Float64(1.0 + Float64(a + Float64(Float64(b * y) / t)))));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), 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[(N[(y / t), $MachinePrecision] * N[(z / N[(1.0 + N[(a + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + 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 30.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 57.2

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in x around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 78.5

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

   7. Applied rewrites 78.5%

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

   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 86.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 83.3

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 83.5

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

   4. Applied rewrites 83.5%

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

   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 96.3

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

   5. Applied rewrites 96.3%

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

Alternative 15: 83.8% accurate, 0.3× speedup

Math

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

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 t) (/ z (fma (/ y t) b (+ a 1.0))))
     (if (<= t_1 INFINITY)
       (/ (fma y (/ z t) x) (fma b (/ y t) (+ 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 / t) * (z / fma((y / t), b, (a + 1.0)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), (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(Float64(y / t) * Float64(z / fma(Float64(y / t), b, Float64(a + 1.0))));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), 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[(N[(y / t), $MachinePrecision] * N[(z / N[(N[(y / t), $MachinePrecision] * b + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + 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 30.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 57.2

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in x around 0

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

      1. times-frac N/A

         \[\leadsto \frac{y}{t} \cdot \color{blue}{\frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{t} \cdot \color{blue}{\frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{y}{t} \cdot \frac{\color{blue}{z}}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{\color{blue}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{\left(1 + a\right) + \color{blue}{\frac{b \cdot y}{t}}} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{\frac{b \cdot y}{t} + \color{blue}{\left(1 + a\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{b \cdot \frac{y}{t} + \left(\color{blue}{1} + a\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{\frac{y}{t} \cdot b + \left(\color{blue}{1} + a\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{\mathsf{fma}\left(\frac{y}{t}, \color{blue}{b}, 1 + a\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{y}{t} \cdot \frac{z}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{t} \cdot \frac{z}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)} \]

      12. lower-+.f64 66.4

          \[\leadsto \frac{y}{t} \cdot \frac{z}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)} \]

   10. Applied rewrites 66.4%

       \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}} \]

   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 86.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      7. lower-/.f64 83.3

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, x\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\left(a + 1\right)} + \frac{y \cdot b}{t}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\left(1 + a\right)} + \frac{y \cdot b}{t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\left(1 + a\right) + \frac{\color{blue}{y \cdot b}}{t}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\left(1 + a\right) + \color{blue}{\frac{y \cdot b}{t}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(1 + a\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\frac{\color{blue}{b \cdot y}}{t} + \left(1 + a\right)} \]

      15. associate-/l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{b \cdot \frac{y}{t}} + \left(1 + a\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \color{blue}{\frac{y}{t}}, 1 + a\right)} \]

      18. lower-+.f64 83.5

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, \color{blue}{1 + a}\right)} \]

   4. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}} \]

   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 96.3

         \[\leadsto \frac{z}{\color{blue}{b}} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + t \cdot \frac{x}{y}}{b}\\ \end{array} \end{array} \]

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 t) (/ z (+ 1.0 a)))
     (if (<= t_1 2e+301)
       (/ x (fma b (/ y t) (+ 1.0 a)))
       (/ (+ z (* t (/ x y))) 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 / t) * (z / (1.0 + a));
	} else if (t_1 <= 2e+301) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else {
		tmp = (z + (t * (x / y))) / 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(Float64(y / t) * Float64(z / Float64(1.0 + a)));
	elseif (t_1 <= 2e+301)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	else
		tmp = Float64(Float64(z + Float64(t * Float64(x / y))) / 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[(N[(y / t), $MachinePrecision] * N[(z / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(t * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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 30.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot z}{t} + x}{\color{blue}{1} + a} \]

      3. associate-/l* N/A

         \[\leadsto \frac{y \cdot \frac{z}{t} + x}{1 + a} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{1} + a} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a} \]

      6. lower-+.f64 43.4

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + \color{blue}{a}} \]

   5. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot z}{\color{blue}{t \cdot \left(1 + a\right)}} \]
   7. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{\color{blue}{1 + a}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{\color{blue}{1 + a}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{\color{blue}{1} + a} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{y}{t} \cdot \frac{z}{1 + \color{blue}{a}} \]

      5. lift-+.f64 50.4

         \[\leadsto \frac{y}{t} \cdot \frac{z}{1 + a} \]

   8. Applied rewrites 50.4%

      \[\leadsto \frac{y}{t} \cdot \color{blue}{\frac{z}{1 + a}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000011e301

   1. Initial program 90.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 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

         \[\leadsto \frac{x}{\color{blue}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{x}{\left(1 + a\right) + \color{blue}{\frac{b \cdot y}{t}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + a\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative N/A

         \[\leadsto \frac{x}{\frac{y \cdot b}{t} + \color{blue}{\left(1 + a\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{x}{b \cdot \frac{y}{t} + \left(\color{blue}{1} + a\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(b, \color{blue}{\frac{y}{t}}, 1 + a\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(b, \frac{y}{\color{blue}{t}}, 1 + a\right)} \]

      9. lower-+.f64 64.6

         \[\leadsto \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)} \]

   5. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}} \]

   if 2.00000000000000011e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 12.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{\left(\left(1 + a\right) \cdot z\right) \cdot t}{b \cdot b}}{y}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      4. lift-*.f64 81.1

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

   8. Applied rewrites 81.1%

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. associate-/l* N/A

         \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]

      5. lower-/.f64 83.3

         \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]

   10. Applied rewrites 83.3%

       \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 86.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)\\ \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, \frac{z}{t\_1}, \frac{x}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b (/ y t) (+ 1.0 a))))
   (if (<= (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) INFINITY)
     (fma (/ y t) (/ z t_1) (/ x t_1))
     (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, (y / t), (1.0 + a));
	double tmp;
	if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= ((double) INFINITY)) {
		tmp = fma((y / t), (z / t_1), (x / t_1));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, Float64(y / t), Float64(1.0 + a))
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t))) <= Inf)
		tmp = fma(Float64(y / t), Float64(z / t_1), Float64(x / t_1));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(y / t), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 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 83.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{\left(a + 1\right)} + \frac{y \cdot b}{t}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\color{blue}{y \cdot b}}{t}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\frac{y \cdot b}{t}}} \]

      9. div-add N/A

         \[\leadsto \color{blue}{\frac{x}{\left(a + 1\right) + \frac{y \cdot b}{t}} + \frac{\frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(1 + a\right)} + \frac{y \cdot b}{t}} + \frac{\frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      11. *-commutative N/A

          \[\leadsto \frac{x}{\left(1 + a\right) + \frac{\color{blue}{b \cdot y}}{t}} + \frac{\frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      12. associate-+r+ N/A

          \[\leadsto \frac{x}{\color{blue}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{\frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      13. +-commutative N/A

          \[\leadsto \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{\frac{y \cdot z}{t}}{\color{blue}{\left(1 + a\right)} + \frac{y \cdot b}{t}} \]

      14. *-commutative N/A

          \[\leadsto \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{\frac{y \cdot z}{t}}{\left(1 + a\right) + \frac{\color{blue}{b \cdot y}}{t}} \]

      15. associate-+r+ N/A

          \[\leadsto \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{\frac{y \cdot z}{t}}{\color{blue}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

   4. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\right)} \]

   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 96.3

         \[\leadsto \frac{z}{\color{blue}{b}} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ t_2 := \frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}\\ \mathbf{if}\;a \leq -56000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (/ z t) x)) (t_2 (/ t_1 (fma b (/ y t) a))))
   (if (<= a -56000.0) t_2 (if (<= a 1.0) (/ t_1 (fma b (/ y t) 1.0)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (z / t), x);
	double t_2 = t_1 / fma(b, (y / t), a);
	double tmp;
	if (a <= -56000.0) {
		tmp = t_2;
	} else if (a <= 1.0) {
		tmp = t_1 / fma(b, (y / t), 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(z / t), x)
	t_2 = Float64(t_1 / fma(b, Float64(y / t), a))
	tmp = 0.0
	if (a <= -56000.0)
		tmp = t_2;
	elseif (a <= 1.0)
		tmp = Float64(t_1 / fma(b, Float64(y / t), 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -56000.0], t$95$2, If[LessEqual[a, 1.0], N[(t$95$1 / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if a < -56000 or 1 < a

   1. Initial program 74.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      7. lower-/.f64 74.4

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, x\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\left(a + 1\right)} + \frac{y \cdot b}{t}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\left(1 + a\right)} + \frac{y \cdot b}{t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\left(1 + a\right) + \frac{\color{blue}{y \cdot b}}{t}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\left(1 + a\right) + \color{blue}{\frac{y \cdot b}{t}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(1 + a\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\frac{\color{blue}{b \cdot y}}{t} + \left(1 + a\right)} \]

      15. associate-/l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{b \cdot \frac{y}{t}} + \left(1 + a\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \color{blue}{\frac{y}{t}}, 1 + a\right)} \]

      18. lower-+.f64 74.9

          \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, \color{blue}{1 + a}\right)} \]

   4. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, \color{blue}{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.1%

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, \color{blue}{a}\right)} \]

   if -56000 < a < 1

   1. Initial program 74.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + \frac{b \cdot y}{t}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + \frac{b \cdot y}{t}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot z}{t} + x}{\color{blue}{1} + \frac{b \cdot y}{t}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{y \cdot \frac{z}{t} + x}{1 + \frac{b \cdot y}{t}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{1} + \frac{b \cdot y}{t}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + \frac{b \cdot y}{t}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\frac{b \cdot y}{t} + \color{blue}{1}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{b \cdot \frac{y}{t} + 1} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \color{blue}{\frac{y}{t}}, 1\right)} \]

      9. lower-/.f64 72.5

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{\color{blue}{t}}, 1\right)} \]

   5. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 60.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 490000000000:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + t \cdot \frac{x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.1e-62)
   (/ (fma t (/ x y) z) b)
   (if (<= y 490000000000.0) (/ x (+ 1.0 a)) (/ (+ z (* t (/ x y))) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.1e-62) {
		tmp = fma(t, (x / y), z) / b;
	} else if (y <= 490000000000.0) {
		tmp = x / (1.0 + a);
	} else {
		tmp = (z + (t * (x / y))) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.1e-62)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (y <= 490000000000.0)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z + Float64(t * Float64(x / y))) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.1e-62], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 490000000000.0], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(t * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1e-62

   1. Initial program 62.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 42.4%

      \[\leadsto \color{blue}{\frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{\left(\left(1 + a\right) \cdot z\right) \cdot t}{b \cdot b}}{y}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      4. lift-*.f64 54.8

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

   8. Applied rewrites 54.8%

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   9. Step-by-step derivation

      1. +-commutative 54.8

         \[\leadsto \frac{\color{blue}{z} + \frac{t \cdot x}{y}}{b} \]

      2. associate-*r/ 54.8

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. +-commutative 54.8

         \[\leadsto \frac{z + \color{blue}{\frac{t \cdot x}{y}}}{b} \]

      4. *-commutative 54.8

         \[\leadsto \frac{z + \frac{\color{blue}{t} \cdot x}{y}}{b} \]

      5. associate-*r/ 54.8

         \[\leadsto \frac{z + \frac{\color{blue}{t \cdot x}}{y}}{b} \]

      6. +-commutative 54.8

         \[\leadsto \frac{z + \frac{t \cdot x}{\color{blue}{y}}}{b} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{t \cdot x}{y} + z}{b} \]

      11. associate-/l* N/A

          \[\leadsto \frac{t \cdot \frac{x}{y} + z}{b} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b} \]

      13. lower-/.f64 57.2

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b} \]

   10. Applied rewrites 57.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}} \]

   if -4.1e-62 < y < 4.9e11

   1. Initial program 94.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 60.8

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   5. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if 4.9e11 < y

   1. Initial program 54.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{\left(\left(1 + a\right) \cdot z\right) \cdot t}{b \cdot b}}{y}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      4. lift-*.f64 60.7

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

   8. Applied rewrites 60.7%

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. associate-/l* N/A

         \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]

      5. lower-/.f64 63.3

         \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]

   10. Applied rewrites 63.3%

       \[\leadsto \frac{z + t \cdot \frac{x}{y}}{b} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 490000000000:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -4.1e-62) t_1 (if (<= y 490000000000.0) (/ x (+ 1.0 a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -4.1e-62) {
		tmp = t_1;
	} else if (y <= 490000000000.0) {
		tmp = x / (1.0 + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -4.1e-62)
		tmp = t_1;
	elseif (y <= 490000000000.0)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -4.1e-62], t$95$1, If[LessEqual[y, 490000000000.0], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1e-62 or 4.9e11 < y

   1. Initial program 58.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{\left(\left(1 + a\right) \cdot z\right) \cdot t}{b \cdot b}}{y}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      4. lift-*.f64 57.5

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

   8. Applied rewrites 57.5%

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
   9. Step-by-step derivation

      1. +-commutative 57.5

         \[\leadsto \frac{\color{blue}{z} + \frac{t \cdot x}{y}}{b} \]

      2. associate-*r/ 57.5

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      3. +-commutative 57.5

         \[\leadsto \frac{z + \color{blue}{\frac{t \cdot x}{y}}}{b} \]

      4. *-commutative 57.5

         \[\leadsto \frac{z + \frac{\color{blue}{t} \cdot x}{y}}{b} \]

      5. associate-*r/ 57.5

         \[\leadsto \frac{z + \frac{\color{blue}{t \cdot x}}{y}}{b} \]

      6. +-commutative 57.5

         \[\leadsto \frac{z + \frac{t \cdot x}{\color{blue}{y}}}{b} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{z + \frac{t \cdot x}{y}}{b} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{t \cdot x}{y} + z}{b} \]

      11. associate-/l* N/A

          \[\leadsto \frac{t \cdot \frac{x}{y} + z}{b} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b} \]

      13. lower-/.f64 59.9

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b} \]

   10. Applied rewrites 59.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}} \]

   if -4.1e-62 < y < 4.9e11

   1. Initial program 94.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 60.8

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   5. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 19.5% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

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
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 74.8%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   4. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   5. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   7. lower-/.f64 73.8

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, x\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\left(a + 1\right)} + \frac{y \cdot b}{t}} \]

   10. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\left(1 + a\right)} + \frac{y \cdot b}{t}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\left(1 + a\right) + \frac{\color{blue}{y \cdot b}}{t}} \]

   12. lift-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\left(1 + a\right) + \color{blue}{\frac{y \cdot b}{t}}} \]

   13. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(1 + a\right)}} \]

   14. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\frac{\color{blue}{b \cdot y}}{t} + \left(1 + a\right)} \]

   15. associate-/l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{b \cdot \frac{y}{t}} + \left(1 + a\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}} \]

   17. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \color{blue}{\frac{y}{t}}, 1 + a\right)} \]

   18. lower-+.f64 74.3

       \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, \color{blue}{1 + a}\right)} \]

4. Applied rewrites 74.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}} \]

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   2. lift-+.f64 41.0

      \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

7. Applied rewrites 41.0%

   \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

8. Taylor expanded in a around 0

   \[\leadsto x \]
9. Step-by-step derivation

10. Applied rewrites 19.5%

    \[\leadsto x \]
11. Add Preprocessing

Developer Target 1: 79.1% 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 2025089 
(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))))