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

Percentage Accurate: 75.3% → 88.4%

Time: 6.2s

Alternatives: 15

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, \frac{y}{t}, a - -1\right)} \cdot \frac{y}{t}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-315}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 10^{+279}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(y, \frac{b}{t}, a - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 (- INFINITY))
     (* (/ z (fma b (/ y t) (- a -1.0))) (/ y t))
     (if (<= t_2 -1e-315)
       t_2
       (if (<= t_2 1e+279) (/ t_1 (fma y (/ b t) (- a -1.0))) (/ z 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 = (z / fma(b, (y / t), (a - -1.0))) * (y / t);
	} else if (t_2 <= -1e-315) {
		tmp = t_2;
	} else if (t_2 <= 1e+279) {
		tmp = t_1 / fma(y, (b / t), (a - -1.0));
	} else {
		tmp = z / 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(z / fma(b, Float64(y / t), Float64(a - -1.0))) * Float64(y / t));
	elseif (t_2 <= -1e-315)
		tmp = t_2;
	elseif (t_2 <= 1e+279)
		tmp = Float64(t_1 / fma(y, Float64(b / t), Float64(a - -1.0)));
	else
		tmp = Float64(z / 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[(z / N[(b * N[(y / t), $MachinePrecision] + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-315], t$95$2, If[LessEqual[t$95$2, 1e+279], N[(t$95$1 / N[(y * N[(b / t), $MachinePrecision] + N[(a - -1.0), $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 75.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 33.4

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

   4. Applied rewrites 33.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 36.2

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

   6. Applied rewrites 36.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.999999985e-316

   1. Initial program 75.3%

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

   if -9.999999985e-316 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000006e279

   1. Initial program 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 74.7

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

   3. Applied rewrites 74.7%

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

   if 1.00000000000000006e279 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.9

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

   4. Applied rewrites 33.9%

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

Alternative 2: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, \frac{y}{t}, a - -1\right)} \cdot \frac{y}{t}\\ \mathbf{elif}\;t\_2 \leq 10^{+279}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(y, \frac{b}{t}, a - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 (- INFINITY))
     (* (/ z (fma b (/ y t) (- a -1.0))) (/ y t))
     (if (<= t_2 1e+279) (/ t_1 (fma y (/ b t) (- a -1.0))) (/ z 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 = (z / fma(b, (y / t), (a - -1.0))) * (y / t);
	} else if (t_2 <= 1e+279) {
		tmp = t_1 / fma(y, (b / t), (a - -1.0));
	} else {
		tmp = z / 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(z / fma(b, Float64(y / t), Float64(a - -1.0))) * Float64(y / t));
	elseif (t_2 <= 1e+279)
		tmp = Float64(t_1 / fma(y, Float64(b / t), Float64(a - -1.0)));
	else
		tmp = Float64(z / 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[(z / N[(b * N[(y / t), $MachinePrecision] + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+279], N[(t$95$1 / N[(y * N[(b / t), $MachinePrecision] + N[(a - -1.0), $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 75.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 33.4

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

   4. Applied rewrites 33.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 36.2

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

   6. Applied rewrites 36.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000006e279

   1. Initial program 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 74.7

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

   3. Applied rewrites 74.7%

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

   if 1.00000000000000006e279 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.9

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

   4. Applied rewrites 33.9%

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

Alternative 3: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \frac{1}{\left(1 + a\right) \cdot t}, \frac{x}{1 + a}\right)\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;a \leq 42000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.5e+15)
   (fma (* z y) (/ 1.0 (* (+ 1.0 a) t)) (/ x (+ 1.0 a)))
   (if (<= a -1.7e-12)
     (/ (+ z (/ (* t x) y)) b)
     (if (<= a 42000000.0)
       (/ (fma y (/ z t) x) (fma b (/ y t) 1.0))
       (/ (+ x (/ (* y z) t)) (- a -1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.5e+15) {
		tmp = fma((z * y), (1.0 / ((1.0 + a) * t)), (x / (1.0 + a)));
	} else if (a <= -1.7e-12) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (a <= 42000000.0) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), 1.0);
	} else {
		tmp = (x + ((y * z) / t)) / (a - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.5e+15)
		tmp = fma(Float64(z * y), Float64(1.0 / Float64(Float64(1.0 + a) * t)), Float64(x / Float64(1.0 + a)));
	elseif (a <= -1.7e-12)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (a <= 42000000.0)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), 1.0));
	else
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(a - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.5e+15], N[(N[(z * y), $MachinePrecision] * N[(1.0 / N[(N[(1.0 + a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.7e-12], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[a, 42000000.0], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.5e15

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 62.5

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 56.0

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

   9. Applied rewrites 56.0%

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

   if -2.5e15 < a < -1.7e-12

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if -1.7e-12 < a < 4.2e7

   1. Initial program 75.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + \frac{b \cdot y}{t}}} \]
   3. 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 45.3

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

   4. Applied rewrites 45.3%

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

   if 4.2e7 < a

   1. Initial program 75.3%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 56.4

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

   4. Applied rewrites 56.4%

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

Alternative 4: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{1 + a} + \frac{y \cdot z}{t \cdot \left(1 + a\right)}\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;a \leq 42000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.5e+15)
   (+ (/ x (+ 1.0 a)) (/ (* y z) (* t (+ 1.0 a))))
   (if (<= a -1.7e-12)
     (/ (+ z (/ (* t x) y)) b)
     (if (<= a 42000000.0)
       (/ (fma y (/ z t) x) (fma b (/ y t) 1.0))
       (/ (+ x (/ (* y z) t)) (- a -1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.5e+15) {
		tmp = (x / (1.0 + a)) + ((y * z) / (t * (1.0 + a)));
	} else if (a <= -1.7e-12) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (a <= 42000000.0) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), 1.0);
	} else {
		tmp = (x + ((y * z) / t)) / (a - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.5e+15)
		tmp = Float64(Float64(x / Float64(1.0 + a)) + Float64(Float64(y * z) / Float64(t * Float64(1.0 + a))));
	elseif (a <= -1.7e-12)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (a <= 42000000.0)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), 1.0));
	else
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(a - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.5e+15], N[(N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / N[(t * N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.7e-12], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[a, 42000000.0], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.5e15

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 56.0

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

   6. Applied rewrites 56.0%

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

   if -2.5e15 < a < -1.7e-12

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if -1.7e-12 < a < 4.2e7

   1. Initial program 75.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + \frac{b \cdot y}{t}}} \]
   3. 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 45.3

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

   4. Applied rewrites 45.3%

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

   if 4.2e7 < a

   1. Initial program 75.3%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 56.4

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

   4. Applied rewrites 56.4%

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

Alternative 5: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\left(1 + a\right) \cdot t}, \frac{x}{1 + a}\right)\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;a \leq 42000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.5e+15)
   (fma y (/ z (* (+ 1.0 a) t)) (/ x (+ 1.0 a)))
   (if (<= a -1.7e-12)
     (/ (+ z (/ (* t x) y)) b)
     (if (<= a 42000000.0)
       (/ (fma y (/ z t) x) (fma b (/ y t) 1.0))
       (/ (+ x (/ (* y z) t)) (- a -1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.5e+15) {
		tmp = fma(y, (z / ((1.0 + a) * t)), (x / (1.0 + a)));
	} else if (a <= -1.7e-12) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (a <= 42000000.0) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), 1.0);
	} else {
		tmp = (x + ((y * z) / t)) / (a - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.5e+15)
		tmp = fma(y, Float64(z / Float64(Float64(1.0 + a) * t)), Float64(x / Float64(1.0 + a)));
	elseif (a <= -1.7e-12)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (a <= 42000000.0)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), 1.0));
	else
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(a - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.5e+15], N[(y * N[(z / N[(N[(1.0 + a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.7e-12], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[a, 42000000.0], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.5e15

   1. Initial program 75.3%

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

   3. Applied rewrites 76.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 63.5

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

   6. Applied rewrites 63.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 56.7

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

   9. Applied rewrites 56.7%

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

   if -2.5e15 < a < -1.7e-12

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if -1.7e-12 < a < 4.2e7

   1. Initial program 75.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + \frac{b \cdot y}{t}}} \]
   3. 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 45.3

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

   4. Applied rewrites 45.3%

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

   if 4.2e7 < a

   1. Initial program 75.3%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 56.4

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

   4. Applied rewrites 56.4%

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

Alternative 6: 68.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{a - -1}\\ \mathbf{if}\;a + 1 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a + 1 \leq 0.999999999999:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;a + 1 \leq 50000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (- a -1.0))))
   (if (<= (+ a 1.0) -5e+16)
     t_1
     (if (<= (+ a 1.0) 0.999999999999)
       (/ (+ z (/ (* t x) y)) b)
       (if (<= (+ a 1.0) 50000000.0)
         (/ (fma y (/ z t) x) (fma b (/ y t) 1.0))
         t_1)))))

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);
	double tmp;
	if ((a + 1.0) <= -5e+16) {
		tmp = t_1;
	} else if ((a + 1.0) <= 0.999999999999) {
		tmp = (z + ((t * x) / y)) / b;
	} else if ((a + 1.0) <= 50000000.0) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(a - -1.0))
	tmp = 0.0
	if (Float64(a + 1.0) <= -5e+16)
		tmp = t_1;
	elseif (Float64(a + 1.0) <= 0.999999999999)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (Float64(a + 1.0) <= 50000000.0)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), 1.0));
	else
		tmp = t_1;
	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[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a + 1.0), $MachinePrecision], -5e+16], t$95$1, If[LessEqual[N[(a + 1.0), $MachinePrecision], 0.999999999999], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[N[(a + 1.0), $MachinePrecision], 50000000.0], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 a #s(literal 1 binary64)) < -5e16 or 5e7 < (+.f64 a #s(literal 1 binary64))

   1. Initial program 75.3%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 56.4

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

   4. Applied rewrites 56.4%

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

   if -5e16 < (+.f64 a #s(literal 1 binary64)) < 0.999999999999000022

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if 0.999999999999000022 < (+.f64 a #s(literal 1 binary64)) < 5e7

   1. Initial program 75.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + \frac{b \cdot y}{t}}} \]
   3. 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 45.3

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

   4. Applied rewrites 45.3%

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

Alternative 7: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(b, y, t \cdot \left(1 + a\right)\right)}\\ \mathbf{elif}\;y \leq -7.4 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a - -1\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ z (/ (* t x) y)) b)))
   (if (<= y -6.2e+173)
     t_1
     (if (<= y -3.5e+61)
       (/ (* z y) (fma b y (* t (+ 1.0 a))))
       (if (<= y -7.4e-162)
         (/ x (fma b (/ y t) (- a -1.0)))
         (if (<= y 4.8e+79) (/ (+ x (/ (* y z) t)) (- a -1.0)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + ((t * x) / y)) / b;
	double tmp;
	if (y <= -6.2e+173) {
		tmp = t_1;
	} else if (y <= -3.5e+61) {
		tmp = (z * y) / fma(b, y, (t * (1.0 + a)));
	} else if (y <= -7.4e-162) {
		tmp = x / fma(b, (y / t), (a - -1.0));
	} else if (y <= 4.8e+79) {
		tmp = (x + ((y * z) / t)) / (a - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + Float64(Float64(t * x) / y)) / b)
	tmp = 0.0
	if (y <= -6.2e+173)
		tmp = t_1;
	elseif (y <= -3.5e+61)
		tmp = Float64(Float64(z * y) / fma(b, y, Float64(t * Float64(1.0 + a))));
	elseif (y <= -7.4e-162)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(a - -1.0)));
	elseif (y <= 4.8e+79)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(a - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -6.2e+173], t$95$1, If[LessEqual[y, -3.5e+61], N[(N[(z * y), $MachinePrecision] / N[(b * y + N[(t * N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -7.4e-162], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+79], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.2e173 or 4.79999999999999971e79 < y

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if -6.2e173 < y < -3.50000000000000018e61

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 33.4

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

   4. Applied rewrites 33.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 41.9

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

   7. Applied rewrites 41.9%

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

   if -3.50000000000000018e61 < y < -7.4000000000000003e-162

   1. Initial program 75.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   3. 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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 54.4

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

   4. Applied rewrites 54.4%

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

   if -7.4000000000000003e-162 < y < 4.79999999999999971e79

   1. Initial program 75.3%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 56.4

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

   4. Applied rewrites 56.4%

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

Alternative 8: 67.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \frac{t\_1}{a - -1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-27}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, x, y \cdot z\right)}{y}}{b}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a - -1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+279}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (/ t_1 (- a -1.0))))
   (if (<= t_2 -1e-27)
     t_3
     (if (<= t_2 -2e-82)
       (/ (/ (fma t x (* y z)) y) b)
       (if (<= t_2 2e-310)
         (/ x (fma b (/ y t) (- a -1.0)))
         (if (<= t_2 1e+279) t_3 (/ z 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 t_3 = t_1 / (a - -1.0);
	double tmp;
	if (t_2 <= -1e-27) {
		tmp = t_3;
	} else if (t_2 <= -2e-82) {
		tmp = (fma(t, x, (y * z)) / y) / b;
	} else if (t_2 <= 2e-310) {
		tmp = x / fma(b, (y / t), (a - -1.0));
	} else if (t_2 <= 1e+279) {
		tmp = t_3;
	} else {
		tmp = z / 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)))
	t_3 = Float64(t_1 / Float64(a - -1.0))
	tmp = 0.0
	if (t_2 <= -1e-27)
		tmp = t_3;
	elseif (t_2 <= -2e-82)
		tmp = Float64(Float64(fma(t, x, Float64(y * z)) / y) / b);
	elseif (t_2 <= 2e-310)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(a - -1.0)));
	elseif (t_2 <= 1e+279)
		tmp = t_3;
	else
		tmp = Float64(z / 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]}, Block[{t$95$3 = N[(t$95$1 / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-27], t$95$3, If[LessEqual[t$95$2, -2e-82], N[(N[(N[(t * x + N[(y * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 2e-310], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+279], t$95$3, 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))) < -1e-27 or 1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000006e279

   1. Initial program 75.3%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 56.4

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

   4. Applied rewrites 56.4%

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

   if -1e-27 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2e-82

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 35.0

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

   9. Applied rewrites 35.0%

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

   if -2e-82 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.999999999999994e-310

   1. Initial program 75.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   3. 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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 54.4

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

   4. Applied rewrites 54.4%

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

   if 1.00000000000000006e279 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.9

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

   4. Applied rewrites 33.9%

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

Alternative 9: 67.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a - -1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, x, y \cdot z\right)}{y}}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a - -1\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+279}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (/ (fma y (/ z t) x) (- a -1.0))))
   (if (<= t_1 -1e-15)
     t_2
     (if (<= t_1 -2e-82)
       (/ (/ (fma t x (* y z)) y) b)
       (if (<= t_1 2e-310)
         (/ x (fma b (/ y t) (- a -1.0)))
         (if (<= t_1 1e+279) 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 = fma(y, (z / t), x) / (a - -1.0);
	double tmp;
	if (t_1 <= -1e-15) {
		tmp = t_2;
	} else if (t_1 <= -2e-82) {
		tmp = (fma(t, x, (y * z)) / y) / b;
	} else if (t_1 <= 2e-310) {
		tmp = x / fma(b, (y / t), (a - -1.0));
	} else if (t_1 <= 1e+279) {
		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(fma(y, Float64(z / t), x) / Float64(a - -1.0))
	tmp = 0.0
	if (t_1 <= -1e-15)
		tmp = t_2;
	elseif (t_1 <= -2e-82)
		tmp = Float64(Float64(fma(t, x, Float64(y * z)) / y) / b);
	elseif (t_1 <= 2e-310)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(a - -1.0)));
	elseif (t_1 <= 1e+279)
		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[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-15], t$95$2, If[LessEqual[t$95$1, -2e-82], N[(N[(N[(t * x + N[(y * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e-310], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+279], t$95$2, 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))) < -1.0000000000000001e-15 or 1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000006e279

   1. Initial program 75.3%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   3. 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. +-commutative N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 56.4

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

   4. Applied rewrites 56.4%

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

   if -1.0000000000000001e-15 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2e-82

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 35.0

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

   9. Applied rewrites 35.0%

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

   if -2e-82 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.999999999999994e-310

   1. Initial program 75.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   3. 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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 54.4

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

   4. Applied rewrites 54.4%

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

   if 1.00000000000000006e279 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.9

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

   4. Applied rewrites 33.9%

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

Alternative 10: 67.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a - -1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a - -1\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+279}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (/ (fma y (/ z t) x) (- a -1.0))))
   (if (<= t_1 -1e-15)
     t_2
     (if (<= t_1 -2e-82)
       (/ (+ z (/ (* t x) y)) b)
       (if (<= t_1 2e-310)
         (/ x (fma b (/ y t) (- a -1.0)))
         (if (<= t_1 1e+279) 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 = fma(y, (z / t), x) / (a - -1.0);
	double tmp;
	if (t_1 <= -1e-15) {
		tmp = t_2;
	} else if (t_1 <= -2e-82) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (t_1 <= 2e-310) {
		tmp = x / fma(b, (y / t), (a - -1.0));
	} else if (t_1 <= 1e+279) {
		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(fma(y, Float64(z / t), x) / Float64(a - -1.0))
	tmp = 0.0
	if (t_1 <= -1e-15)
		tmp = t_2;
	elseif (t_1 <= -2e-82)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (t_1 <= 2e-310)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(a - -1.0)));
	elseif (t_1 <= 1e+279)
		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[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-15], t$95$2, If[LessEqual[t$95$1, -2e-82], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e-310], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+279], t$95$2, 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))) < -1.0000000000000001e-15 or 1.999999999999994e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000006e279

   1. Initial program 75.3%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   3. 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. +-commutative N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 56.4

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

   4. Applied rewrites 56.4%

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

   if -1.0000000000000001e-15 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2e-82

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if -2e-82 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.999999999999994e-310

   1. Initial program 75.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   3. 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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 54.4

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

   4. Applied rewrites 54.4%

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

   if 1.00000000000000006e279 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.9

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

   4. Applied rewrites 33.9%

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

Alternative 11: 65.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ z (/ (* t x) y)) b)))
   (if (<= y -4.3e+42)
     t_1
     (if (<= y 1.3e+97) (/ x (fma b (/ y t) (- a -1.0))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + ((t * x) / y)) / b;
	double tmp;
	if (y <= -4.3e+42) {
		tmp = t_1;
	} else if (y <= 1.3e+97) {
		tmp = x / fma(b, (y / t), (a - -1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + Float64(Float64(t * x) / y)) / b)
	tmp = 0.0
	if (y <= -4.3e+42)
		tmp = t_1;
	elseif (y <= 1.3e+97)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(a - -1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -4.3e+42], t$95$1, If[LessEqual[y, 1.3e+97], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2999999999999998e42 or 1.3e97 < y

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if -4.2999999999999998e42 < y < 1.3e97

   1. Initial program 75.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   3. 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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 54.4

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

   4. Applied rewrites 54.4%

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

Alternative 12: 59.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00016:\\ \;\;\;\;\frac{x}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ z (/ (* t x) y)) b)))
   (if (<= y -2.2e+41) t_1 (if (<= y 0.00016) (/ x (- a -1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + ((t * x) / y)) / b;
	double tmp;
	if (y <= -2.2e+41) {
		tmp = t_1;
	} else if (y <= 0.00016) {
		tmp = x / (a - -1.0);
	} 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 = (z + ((t * x) / y)) / b
    if (y <= (-2.2d+41)) then
        tmp = t_1
    else if (y <= 0.00016d0) then
        tmp = x / (a - (-1.0d0))
    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 = (z + ((t * x) / y)) / b;
	double tmp;
	if (y <= -2.2e+41) {
		tmp = t_1;
	} else if (y <= 0.00016) {
		tmp = x / (a - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z + ((t * x) / y)) / b
	tmp = 0
	if y <= -2.2e+41:
		tmp = t_1
	elif y <= 0.00016:
		tmp = x / (a - -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + Float64(Float64(t * x) / y)) / b)
	tmp = 0.0
	if (y <= -2.2e+41)
		tmp = t_1;
	elseif (y <= 0.00016)
		tmp = Float64(x / Float64(a - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z + ((t * x) / y)) / b;
	tmp = 0.0;
	if (y <= -2.2e+41)
		tmp = t_1;
	elseif (y <= 0.00016)
		tmp = x / (a - -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -2.2e+41], t$95$1, If[LessEqual[y, 0.00016], N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999999e41 or 1.60000000000000013e-4 < y

   1. Initial program 75.3%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      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. lower-*.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if -2.1999999999999999e41 < y < 1.60000000000000013e-4

   1. Initial program 75.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{x}{a - -1} \]

      5. lower--.f64 42.3

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

   4. Applied rewrites 42.3%

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

Alternative 13: 56.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.3e+42) (/ z b) (if (<= y 5e+79) (/ x (- a -1.0)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.3e+42) {
		tmp = z / b;
	} else if (y <= 5e+79) {
		tmp = x / (a - -1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-4.3d+42)) then
        tmp = z / b
    else if (y <= 5d+79) then
        tmp = x / (a - (-1.0d0))
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.3e+42) {
		tmp = z / b;
	} else if (y <= 5e+79) {
		tmp = x / (a - -1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4.3e+42:
		tmp = z / b
	elif y <= 5e+79:
		tmp = x / (a - -1.0)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.3e+42)
		tmp = Float64(z / b);
	elseif (y <= 5e+79)
		tmp = Float64(x / Float64(a - -1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4.3e+42)
		tmp = z / b;
	elseif (y <= 5e+79)
		tmp = x / (a - -1.0);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.3e+42], N[(z / b), $MachinePrecision], If[LessEqual[y, 5e+79], N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2999999999999998e42 or 5e79 < y

   1. Initial program 75.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.9

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

   4. Applied rewrites 33.9%

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

   if -4.2999999999999998e42 < y < 5e79

   1. Initial program 75.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{x}{a - -1} \]

      5. lower--.f64 42.3

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

   4. Applied rewrites 42.3%

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

Alternative 14: 42.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 105000000:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.5e+15) (/ x a) (if (<= a 105000000.0) (/ z b) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.5e+15) {
		tmp = x / a;
	} else if (a <= 105000000.0) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.5d+15)) then
        tmp = x / a
    else if (a <= 105000000.0d0) then
        tmp = z / b
    else
        tmp = x / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.5e+15) {
		tmp = x / a;
	} else if (a <= 105000000.0) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -2.5e+15:
		tmp = x / a
	elif a <= 105000000.0:
		tmp = z / b
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.5e+15)
		tmp = Float64(x / a);
	elseif (a <= 105000000.0)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -2.5e+15)
		tmp = x / a;
	elseif (a <= 105000000.0)
		tmp = z / b;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.5e+15], N[(x / a), $MachinePrecision], If[LessEqual[a, 105000000.0], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5e15 or 1.05e8 < a

   1. Initial program 75.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{x}{a - -1} \]

      5. lower--.f64 42.3

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

   4. Applied rewrites 42.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 19.3

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

   7. Applied rewrites 19.3%

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

   8. Taylor expanded in a around inf

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

      1. lower-/.f64 25.6

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

   10. Applied rewrites 25.6%

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

   if -2.5e15 < a < 1.05e8

   1. Initial program 75.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.9

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

   4. Applied rewrites 33.9%

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

Alternative 15: 25.6% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

2. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{x}{a - -1} \]

   5. lower--.f64 42.3

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

4. Applied rewrites 42.3%

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

5. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 19.3

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

7. Applied rewrites 19.3%

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

8. Taylor expanded in a around inf

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

   1. lower-/.f64 25.6

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

10. Applied rewrites 25.6%

    \[\leadsto \frac{x}{\color{blue}{a}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64
  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))