Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.1% → 88.4%

Time: 5.2s

Alternatives: 18

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

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))) / (y + (z * (b - y)))
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))) / (y + (z * (b - y)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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))) / (y + (z * (b - y)))
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))) / (y + (z * (b - y)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{1}{b - y}, t - a, \frac{x}{z \cdot \left(b - y\right)} \cdot y\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ 1.0 (- b y)) (- t a) (* (/ x (* z (- b y))) y))))
   (if (<= z -2.4e+33)
     t_1
     (if (<= z 6.2e+26) (/ (fma (- t a) z (* y x)) (fma (- b y) z y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((1.0 / (b - y)), (t - a), ((x / (z * (b - y))) * y));
	double tmp;
	if (z <= -2.4e+33) {
		tmp = t_1;
	} else if (z <= 6.2e+26) {
		tmp = fma((t - a), z, (y * x)) / fma((b - y), z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(1.0 / Float64(b - y)), Float64(t - a), Float64(Float64(x / Float64(z * Float64(b - y))) * y))
	tmp = 0.0
	if (z <= -2.4e+33)
		tmp = t_1;
	elseif (z <= 6.2e+26)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(b - y), z, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision] + N[(N[(x / N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e+33], t$95$1, If[LessEqual[z, 6.2e+26], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e33 or 6.1999999999999999e26 < z

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 73.4%

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 59.1%

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

   9. Applied rewrites 59.1%

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

   if -2.4e33 < z < 6.1999999999999999e26

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.1%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.1%

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

   3. Applied rewrites 66.1%

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

Alternative 2: 87.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{1}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_4 := \mathsf{fma}\left(b - y, z, y\right)\\ t_5 := \frac{x}{t\_4} \cdot y\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{t\_4}, z, t\_5\right)\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-314}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t - a, \frac{\mathsf{fma}\left(-1, \frac{x \cdot y}{z \cdot {\left(b - y\right)}^{2}}, \frac{x}{b - y}\right)}{z} \cdot y\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(-a, z, y \cdot x\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t - a, t\_5\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ 1.0 (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (fma (- b y) z y))
        (t_5 (* (/ x t_4) y)))
   (if (<= t_3 (- INFINITY))
     (fma (/ (- t a) t_4) z t_5)
     (if (<= t_3 -1e-314)
       (/ (fma (- t a) z (* y x)) t_4)
       (if (<= t_3 0.0)
         (fma
          t_2
          (- t a)
          (*
           (/ (fma -1.0 (/ (* x y) (* z (pow (- b y) 2.0))) (/ x (- b y))) z)
           y))
         (if (<= t_3 INFINITY)
           (/ (fma z t (fma (- a) z (* y x))) t_1)
           (fma t_2 (- t a) t_5)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = 1.0 / (b - y);
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = fma((b - y), z, y);
	double t_5 = (x / t_4) * y;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(((t - a) / t_4), z, t_5);
	} else if (t_3 <= -1e-314) {
		tmp = fma((t - a), z, (y * x)) / t_4;
	} else if (t_3 <= 0.0) {
		tmp = fma(t_2, (t - a), ((fma(-1.0, ((x * y) / (z * pow((b - y), 2.0))), (x / (b - y))) / z) * y));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma(z, t, fma(-a, z, (y * x))) / t_1;
	} else {
		tmp = fma(t_2, (t - a), t_5);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(1.0 / Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = fma(Float64(b - y), z, y)
	t_5 = Float64(Float64(x / t_4) * y)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(Float64(Float64(t - a) / t_4), z, t_5);
	elseif (t_3 <= -1e-314)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / t_4);
	elseif (t_3 <= 0.0)
		tmp = fma(t_2, Float64(t - a), Float64(Float64(fma(-1.0, Float64(Float64(x * y) / Float64(z * (Float64(b - y) ^ 2.0))), Float64(x / Float64(b - y))) / z) * y));
	elseif (t_3 <= Inf)
		tmp = Float64(fma(z, t, fma(Float64(-a), z, Float64(y * x))) / t_1);
	else
		tmp = fma(t_2, Float64(t - a), t_5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x / t$95$4), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(t - a), $MachinePrecision] / t$95$4), $MachinePrecision] * z + t$95$5), $MachinePrecision], If[LessEqual[t$95$3, -1e-314], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(t$95$2 * N[(t - a), $MachinePrecision] + N[(N[(N[(-1.0 * N[(N[(x * y), $MachinePrecision] / N[(z * N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(z * t + N[((-a) * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$2 * N[(t - a), $MachinePrecision] + t$95$5), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 69.4%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999996e-315

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.1%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.1%

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

   3. Applied rewrites 66.1%

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

   if -9.9999999996e-315 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 73.4%

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 52.6%

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

   9. Applied rewrites 52.6%

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

   if -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 66.0%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

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

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 73.4%

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

   6. Applied rewrites 73.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{b - y}}, t - a, \frac{x}{\mathsf{fma}\left(b - y, z, y\right)} \cdot y\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_3 := \mathsf{fma}\left(b - y, z, y\right)\\ t_4 := \frac{x}{t\_3} \cdot y\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t\_3}, t - a, t\_4\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{x \cdot y}{b - y}, \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}\right)}{z}, \frac{t}{b - y}\right) - \frac{a}{b - y}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(-a, z, y \cdot x\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{b - y}, t - a, t\_4\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_3 (fma (- b y) z y))
        (t_4 (* (/ x t_3) y)))
   (if (<= t_2 -1e-234)
     (fma (/ z t_3) (- t a) t_4)
     (if (<= t_2 0.0)
       (-
        (fma
         -1.0
         (/
          (fma -1.0 (/ (* x y) (- b y)) (/ (* y (- t a)) (pow (- b y) 2.0)))
          z)
         (/ t (- b y)))
        (/ a (- b y)))
       (if (<= t_2 INFINITY)
         (/ (fma z t (fma (- a) z (* y x))) t_1)
         (fma (/ 1.0 (- b y)) (- t a) t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = ((x * y) + (z * (t - a))) / t_1;
	double t_3 = fma((b - y), z, y);
	double t_4 = (x / t_3) * y;
	double tmp;
	if (t_2 <= -1e-234) {
		tmp = fma((z / t_3), (t - a), t_4);
	} else if (t_2 <= 0.0) {
		tmp = fma(-1.0, (fma(-1.0, ((x * y) / (b - y)), ((y * (t - a)) / pow((b - y), 2.0))) / z), (t / (b - y))) - (a / (b - y));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma(z, t, fma(-a, z, (y * x))) / t_1;
	} else {
		tmp = fma((1.0 / (b - y)), (t - a), t_4);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_3 = fma(Float64(b - y), z, y)
	t_4 = Float64(Float64(x / t_3) * y)
	tmp = 0.0
	if (t_2 <= -1e-234)
		tmp = fma(Float64(z / t_3), Float64(t - a), t_4);
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(-1.0, Float64(fma(-1.0, Float64(Float64(x * y) / Float64(b - y)), Float64(Float64(y * Float64(t - a)) / (Float64(b - y) ^ 2.0))) / z), Float64(t / Float64(b - y))) - Float64(a / Float64(b - y)));
	elseif (t_2 <= Inf)
		tmp = Float64(fma(z, t, fma(Float64(-a), z, Float64(y * x))) / t_1);
	else
		tmp = fma(Float64(1.0 / Float64(b - y)), Float64(t - a), t_4);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x / t$95$3), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-234], N[(N[(z / t$95$3), $MachinePrecision] * N[(t - a), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(-1.0 * N[(N[(-1.0 * N[(N[(x * y), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(z * t + N[((-a) * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(1.0 / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision] + t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999996e-235

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

   3. Applied rewrites 72.1%

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

   if -9.9999999999999996e-235 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in z around -inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 47.8%

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

   if -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 66.0%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

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

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 73.4%

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

   6. Applied rewrites 73.4%

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

Alternative 4: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.86 \cdot 10^{+24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.6e+61)
     t_1
     (if (<= z 1.86e+24) (/ (fma (- t a) z (* y x)) (fma (- b y) z y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.6e+61) {
		tmp = t_1;
	} else if (z <= 1.86e+24) {
		tmp = fma((t - a), z, (y * x)) / fma((b - y), z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.6e+61)
		tmp = t_1;
	elseif (z <= 1.86e+24)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(b - y), z, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.6e+61], t$95$1, If[LessEqual[z, 1.86e+24], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5999999999999999e61 or 1.86e24 < z

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.7%

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -1.5999999999999999e61 < z < 1.86e24

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.1%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.1%

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

   3. Applied rewrites 66.1%

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

Alternative 5: 83.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_3 := \mathsf{fma}\left(b - y, z, y\right)\\ t_4 := \frac{x}{t\_3} \cdot y\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{t\_3}, z, t\_4\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(-a, z, y \cdot x\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{b - y}, t - a, t\_4\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_3 (fma (- b y) z y))
        (t_4 (* (/ x t_3) y)))
   (if (<= t_2 (- INFINITY))
     (fma (/ (- t a) t_3) z t_4)
     (if (<= t_2 INFINITY)
       (/ (fma z t (fma (- a) z (* y x))) t_1)
       (fma (/ 1.0 (- b y)) (- t a) t_4)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = ((x * y) + (z * (t - a))) / t_1;
	double t_3 = fma((b - y), z, y);
	double t_4 = (x / t_3) * y;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(((t - a) / t_3), z, t_4);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma(z, t, fma(-a, z, (y * x))) / t_1;
	} else {
		tmp = fma((1.0 / (b - y)), (t - a), t_4);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_3 = fma(Float64(b - y), z, y)
	t_4 = Float64(Float64(x / t_3) * y)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(Float64(t - a) / t_3), z, t_4);
	elseif (t_2 <= Inf)
		tmp = Float64(fma(z, t, fma(Float64(-a), z, Float64(y * x))) / t_1);
	else
		tmp = fma(Float64(1.0 / Float64(b - y)), Float64(t - a), t_4);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x / t$95$3), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] * z + t$95$4), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(z * t + N[((-a) * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(1.0 / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision] + t$95$4), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 69.4%

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

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

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 66.0%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

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

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 73.4%

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

   6. Applied rewrites 73.4%

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

Alternative 6: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{t - a}{\left(b - y\right) + \frac{y}{z}}}}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.2e+38)
   (/ (- t a) (- b y))
   (if (<= z 3.3e-18)
     (/ (fma (- t a) z (* y x)) (fma b z y))
     (/ 1.0 (/ 1.0 (/ (- t a) (+ (- b y) (/ y z))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.2e+38) {
		tmp = (t - a) / (b - y);
	} else if (z <= 3.3e-18) {
		tmp = fma((t - a), z, (y * x)) / fma(b, z, y);
	} else {
		tmp = 1.0 / (1.0 / ((t - a) / ((b - y) + (y / z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.2e+38)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= 3.3e-18)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(b, z, y));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(t - a) / Float64(Float64(b - y) + Float64(y / z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.2e+38], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e-18], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(b * z + y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.20000000000000009e38

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.7%

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -1.20000000000000009e38 < z < 3.3000000000000002e-18

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.1%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.1%

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

   3. Applied rewrites 66.1%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 56.4%

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

   if 3.3000000000000002e-18 < z

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 66.0%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 66.0%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 66.0%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 41.0%

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

   6. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. add-to-fraction-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 60.9%

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

   8. Applied rewrites 60.9%

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

Alternative 7: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(b - y\right) + \frac{y}{z}}{t - a}}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.2e+38)
   (/ (- t a) (- b y))
   (if (<= z 3.3e-18)
     (/ (fma (- t a) z (* y x)) (fma b z y))
     (/ 1.0 (/ (+ (- b y) (/ y z)) (- t a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.2e+38) {
		tmp = (t - a) / (b - y);
	} else if (z <= 3.3e-18) {
		tmp = fma((t - a), z, (y * x)) / fma(b, z, y);
	} else {
		tmp = 1.0 / (((b - y) + (y / z)) / (t - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.2e+38)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= 3.3e-18)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(b, z, y));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b - y) + Float64(y / z)) / Float64(t - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.2e+38], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e-18], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(b * z + y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(b - y), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.20000000000000009e38

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.7%

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -1.20000000000000009e38 < z < 3.3000000000000002e-18

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.1%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.1%

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

   3. Applied rewrites 66.1%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 56.4%

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

   if 3.3000000000000002e-18 < z

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 66.0%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 66.0%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 66.0%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 41.0%

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

   6. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-to-fraction-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 60.8%

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

   8. Applied rewrites 60.8%

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

Alternative 8: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, \left(t - a\right) \cdot z\right)}{\mathsf{fma}\left(b, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(b - y\right) + \frac{y}{z}}{t - a}}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.2e+38)
   (/ (- t a) (- b y))
   (if (<= z 3.3e-18)
     (/ (fma x y (* (- t a) z)) (fma b z y))
     (/ 1.0 (/ (+ (- b y) (/ y z)) (- t a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.2e+38) {
		tmp = (t - a) / (b - y);
	} else if (z <= 3.3e-18) {
		tmp = fma(x, y, ((t - a) * z)) / fma(b, z, y);
	} else {
		tmp = 1.0 / (((b - y) + (y / z)) / (t - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.2e+38)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= 3.3e-18)
		tmp = Float64(fma(x, y, Float64(Float64(t - a) * z)) / fma(b, z, y));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b - y) + Float64(y / z)) / Float64(t - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.2e+38], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e-18], N[(N[(x * y + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(b * z + y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(b - y), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.20000000000000009e38

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.7%

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -1.20000000000000009e38 < z < 3.3000000000000002e-18

   1. Initial program 66.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 56.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 56.4%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.4%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 56.4%

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

   6. Applied rewrites 56.4%

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

   if 3.3000000000000002e-18 < z

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 66.0%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 66.0%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 66.0%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 41.0%

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

   6. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-to-fraction-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 60.8%

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

   8. Applied rewrites 60.8%

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

Alternative 9: 82.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_3 (fma (/ 1.0 (- b y)) (- t a) (* (/ x (fma (- b y) z y)) y))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 INFINITY) (/ (fma z t (fma (- a) z (* y x))) t_1) t_3))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 73.4%

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

   6. Applied rewrites 73.4%

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

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

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 66.0%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

Alternative 10: 81.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 73.4%

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 39.0%

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

   9. Applied rewrites 39.0%

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

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

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.1%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.1%

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

   3. Applied rewrites 66.1%

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

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

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 66.0%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 66.0%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 66.0%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 41.0%

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

   6. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. add-to-fraction-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 60.9%

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

   8. Applied rewrites 60.9%

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

Alternative 11: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \frac{1}{\frac{\left(b - y\right) + \frac{y}{z}}{t - a}}\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{y + z \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ 1.0 (/ (+ (- b y) (/ y z)) (- t a)))))
   (if (<= z -5.2e-31)
     t_1
     (if (<= z 1.6e-41) (/ (fma t z (* x y)) (+ y (* z b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 / (((b - y) + (y / z)) / (t - a));
	double tmp;
	if (z <= -5.2e-31) {
		tmp = t_1;
	} else if (z <= 1.6e-41) {
		tmp = fma(t, z, (x * y)) / (y + (z * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 / Float64(Float64(Float64(b - y) + Float64(y / z)) / Float64(t - a)))
	tmp = 0.0
	if (z <= -5.2e-31)
		tmp = t_1;
	elseif (z <= 1.6e-41)
		tmp = Float64(fma(t, z, Float64(x * y)) / Float64(y + Float64(z * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 / N[(N[(N[(b - y), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e-31], t$95$1, If[LessEqual[z, 1.6e-41], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.19999999999999991e-31 or 1.60000000000000006e-41 < z

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 66.0%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 66.0%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 66.0%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 41.0%

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

   6. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-to-fraction-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 60.8%

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

   8. Applied rewrites 60.8%

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

   if -5.19999999999999991e-31 < z < 1.60000000000000006e-41

   1. Initial program 66.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 47.7%

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 41.8%

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

Alternative 12: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{y + z \cdot b}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.8e-10)
     t_1
     (if (<= z 1.6e-41)
       (/ (fma t z (* x y)) (+ y (* z b)))
       (if (<= z 1.5e+15) (/ (* z (- t a)) (fma (- b y) z y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.8e-10) {
		tmp = t_1;
	} else if (z <= 1.6e-41) {
		tmp = fma(t, z, (x * y)) / (y + (z * b));
	} else if (z <= 1.5e+15) {
		tmp = (z * (t - a)) / fma((b - y), z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.8e-10)
		tmp = t_1;
	elseif (z <= 1.6e-41)
		tmp = Float64(fma(t, z, Float64(x * y)) / Float64(y + Float64(z * b)));
	elseif (z <= 1.5e+15)
		tmp = Float64(Float64(z * Float64(t - a)) / fma(Float64(b - y), z, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e-10], t$95$1, If[LessEqual[z, 1.6e-41], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e+15], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.80000000000000015e-10 or 1.5e15 < z

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.7%

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -2.80000000000000015e-10 < z < 1.60000000000000006e-41

   1. Initial program 66.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 47.7%

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 41.8%

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

   if 1.60000000000000006e-41 < z < 1.5e15

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.1%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.1%

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

   3. Applied rewrites 66.1%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 41.1%

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

   6. Applied rewrites 41.1%

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

Alternative 13: 66.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{-159}:\\ \;\;\;\;-1 \cdot \frac{x}{z - 1}\\ \mathbf{elif}\;z \leq 65000000000:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.6e-50)
     t_1
     (if (<= z 7.3e-159)
       (* -1.0 (/ x (- z 1.0)))
       (if (<= z 65000000000.0) (/ (* z (- t a)) (fma (- b y) z y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.6e-50) {
		tmp = t_1;
	} else if (z <= 7.3e-159) {
		tmp = -1.0 * (x / (z - 1.0));
	} else if (z <= 65000000000.0) {
		tmp = (z * (t - a)) / fma((b - y), z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.6e-50)
		tmp = t_1;
	elseif (z <= 7.3e-159)
		tmp = Float64(-1.0 * Float64(x / Float64(z - 1.0)));
	elseif (z <= 65000000000.0)
		tmp = Float64(Float64(z * Float64(t - a)) / fma(Float64(b - y), z, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-50], t$95$1, If[LessEqual[z, 7.3e-159], N[(-1.0 * N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 65000000000.0], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.59999999999999979e-50 or 6.5e10 < z

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.7%

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.59999999999999979e-50 < z < 7.2999999999999996e-159

   1. Initial program 66.1%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 33.4%

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

   4. Applied rewrites 33.4%

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

   if 7.2999999999999996e-159 < z < 6.5e10

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.1%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.1%

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

   3. Applied rewrites 66.1%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 41.1%

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

   6. Applied rewrites 41.1%

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

Alternative 14: 64.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;-1 \cdot \frac{x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.6e-50) t_1 (if (<= z 4.2e-42) (* -1.0 (/ x (- z 1.0))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.6e-50) {
		tmp = t_1;
	} else if (z <= 4.2e-42) {
		tmp = -1.0 * (x / (z - 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 = (t - a) / (b - y)
    if (z <= (-3.6d-50)) then
        tmp = t_1
    else if (z <= 4.2d-42) then
        tmp = (-1.0d0) * (x / (z - 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 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.6e-50) {
		tmp = t_1;
	} else if (z <= 4.2e-42) {
		tmp = -1.0 * (x / (z - 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -3.6e-50:
		tmp = t_1
	elif z <= 4.2e-42:
		tmp = -1.0 * (x / (z - 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.6e-50)
		tmp = t_1;
	elseif (z <= 4.2e-42)
		tmp = Float64(-1.0 * Float64(x / Float64(z - 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -3.6e-50)
		tmp = t_1;
	elseif (z <= 4.2e-42)
		tmp = -1.0 * (x / (z - 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[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-50], t$95$1, If[LessEqual[z, 4.2e-42], N[(-1.0 * N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999979e-50 or 4.20000000000000013e-42 < z

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.7%

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.59999999999999979e-50 < z < 4.20000000000000013e-42

   1. Initial program 66.1%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 33.4%

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

   4. Applied rewrites 33.4%

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

Alternative 15: 64.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.6e-50) t_1 (if (<= z 4.2e-42) (/ 1.0 (/ 1.0 x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.6e-50) {
		tmp = t_1;
	} else if (z <= 4.2e-42) {
		tmp = 1.0 / (1.0 / x);
	} 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 = (t - a) / (b - y)
    if (z <= (-3.6d-50)) then
        tmp = t_1
    else if (z <= 4.2d-42) then
        tmp = 1.0d0 / (1.0d0 / x)
    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 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.6e-50) {
		tmp = t_1;
	} else if (z <= 4.2e-42) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -3.6e-50:
		tmp = t_1
	elif z <= 4.2e-42:
		tmp = 1.0 / (1.0 / x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.6e-50)
		tmp = t_1;
	elseif (z <= 4.2e-42)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -3.6e-50)
		tmp = t_1;
	elseif (z <= 4.2e-42)
		tmp = 1.0 / (1.0 / x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-50], t$95$1, If[LessEqual[z, 4.2e-42], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999979e-50 or 4.20000000000000013e-42 < z

   1. Initial program 66.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.7%

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.59999999999999979e-50 < z < 4.20000000000000013e-42

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 66.0%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 66.0%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 66.0%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 25.5%

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

   6. Applied rewrites 25.5%

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

Alternative 16: 48.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) b)))
   (if (<= z -7.8e-50) t_1 (if (<= z 4.5e-42) (/ 1.0 (/ 1.0 x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / b;
	double tmp;
	if (z <= -7.8e-50) {
		tmp = t_1;
	} else if (z <= 4.5e-42) {
		tmp = 1.0 / (1.0 / x);
	} 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 = (t - a) / b
    if (z <= (-7.8d-50)) then
        tmp = t_1
    else if (z <= 4.5d-42) then
        tmp = 1.0d0 / (1.0d0 / x)
    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 = (t - a) / b;
	double tmp;
	if (z <= -7.8e-50) {
		tmp = t_1;
	} else if (z <= 4.5e-42) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / b
	tmp = 0
	if z <= -7.8e-50:
		tmp = t_1
	elif z <= 4.5e-42:
		tmp = 1.0 / (1.0 / x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / b)
	tmp = 0.0
	if (z <= -7.8e-50)
		tmp = t_1;
	elseif (z <= 4.5e-42)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / b;
	tmp = 0.0;
	if (z <= -7.8e-50)
		tmp = t_1;
	elseif (z <= 4.5e-42)
		tmp = 1.0 / (1.0 / x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[z, -7.8e-50], t$95$1, If[LessEqual[z, 4.5e-42], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.80000000000000042e-50 or 4.5e-42 < z

   1. Initial program 66.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b}} \]

      2. lower--.f64 35.8%

         \[\leadsto \frac{t - a}{b} \]

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]

   if -7.80000000000000042e-50 < z < 4.5e-42

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 66.0%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 66.0%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 66.0%

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

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{x \cdot y}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}} \]

      17. lower-*.f64 66.0%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}} \]

   3. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, y \cdot x\right)}}} \]

   4. Taylor expanded in z around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 25.5%

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x}}} \]

   6. Applied rewrites 25.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 35.8% accurate, 3.4× speedup

Math

\[\frac{t - a}{b} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ (- t a) b))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (t - a) / b;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (t - a) / b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (t - a) / b;
}

Python

def code(x, y, z, t, a, b):
	return (t - a) / b

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(t - a) / b)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (t - a) / b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{t - a}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{t - a}{\color{blue}{b}} \]

   2. lower--.f64 35.8%

      \[\leadsto \frac{t - a}{b} \]

4. Applied rewrites 35.8%

   \[\leadsto \color{blue}{\frac{t - a}{b}} \]
5. Add Preprocessing

Alternative 18: 20.6% accurate, 5.5× speedup

Math

\[\frac{t}{b} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ t b))

C

double code(double x, double y, double z, double t, double a, double b) {
	return t / b;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = t / b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return t / b;
}

Python

def code(x, y, z, t, a, b):
	return t / b

Julia

function code(x, y, z, t, a, b)
	return Float64(t / b)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = t / b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(t / b), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{t - a}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{t - a}{\color{blue}{b}} \]

   2. lower--.f64 35.8%

      \[\leadsto \frac{t - a}{b} \]

4. Applied rewrites 35.8%

   \[\leadsto \color{blue}{\frac{t - a}{b}} \]

5. Taylor expanded in t around inf

   \[\leadsto \frac{t}{\color{blue}{b}} \]
6. Step-by-step derivation

   1. lower-/.f64 20.6%

      \[\leadsto \frac{t}{b} \]

7. Applied rewrites 20.6%

   \[\leadsto \frac{t}{\color{blue}{b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025184 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64
  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))