Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 67.0% → 93.5%

Time: 10.8s

Alternatives: 16

Speedup: 1.3×

Specification

Math

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

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: 67.0% accurate, 1.0× speedup

Math

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

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: 93.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+59} \lor \neg \left(z \leq 1400000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{t - a}{{\left(b - y\right)}^{2}}, \frac{a}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{t\_1}, \frac{\left(t - a\right) \cdot z}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)))
   (if (or (<= z -1.95e+59) (not (<= z 1400000000.0)))
     (-
      (fma (/ x z) (/ y (- b y)) (/ t (- b y)))
      (fma (/ y z) (/ (- t a) (pow (- b y) 2.0)) (/ a (- b y))))
     (fma x (/ y t_1) (/ (* (- t a) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double tmp;
	if ((z <= -1.95e+59) || !(z <= 1400000000.0)) {
		tmp = fma((x / z), (y / (b - y)), (t / (b - y))) - fma((y / z), ((t - a) / pow((b - y), 2.0)), (a / (b - y)));
	} else {
		tmp = fma(x, (y / t_1), (((t - a) * z) / t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	tmp = 0.0
	if ((z <= -1.95e+59) || !(z <= 1400000000.0))
		tmp = Float64(fma(Float64(x / z), Float64(y / Float64(b - y)), Float64(t / Float64(b - y))) - fma(Float64(y / z), Float64(Float64(t - a) / (Float64(b - y) ^ 2.0)), Float64(a / Float64(b - y))));
	else
		tmp = fma(x, Float64(y / t_1), Float64(Float64(Float64(t - a) * z) / t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, If[Or[LessEqual[z, -1.95e+59], N[Not[LessEqual[z, 1400000000.0]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / z), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / t$95$1), $MachinePrecision] + N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95000000000000011e59 or 1.4e9 < z

   1. Initial program 36.3%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 90.2%

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

   if -1.95000000000000011e59 < z < 1.4e9

   1. Initial program 90.1%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 97.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+59} \lor \neg \left(z \leq 1400000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{t - a}{{\left(b - y\right)}^{2}}, \frac{a}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-269}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{b}}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))))
   (if (<= t_1 (- INFINITY))
     (* (fma (/ z x) (/ (- t a) (fma (- b y) z y)) 1.0) x)
     (if (<= t_1 -1e-269)
       t_1
       (if (<= t_1 0.0)
         (/ (/ (fma (- t a) z (* y x)) b) z)
         (if (<= t_1 5e+307) t_1 (/ (- t a) (- b y))))))))

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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((z / x), ((t - a) / fma((b - y), z, y)), 1.0) * x;
	} else if (t_1 <= -1e-269) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (fma((t - a), z, (y * x)) / b) / z;
	} else if (t_1 <= 5e+307) {
		tmp = t_1;
	} else {
		tmp = (t - a) / (b - y);
	}
	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))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(z / x), Float64(Float64(t - a) / fma(Float64(b - y), z, y)), 1.0) * x);
	elseif (t_1 <= -1e-269)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(Float64(t - a), z, Float64(y * x)) / b) / z);
	elseif (t_1 <= 5e+307)
		tmp = t_1;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(z / x), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, -1e-269], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+307], t$95$1, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $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)))) < -inf.0

   1. Initial program 28.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999996e-270 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5e307

   1. Initial program 99.6%

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

   if -9.9999999999999996e-270 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 19.3%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   if 5e307 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 14.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 72.0

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

   5. Applied rewrites 72.0%

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

Alternative 3: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-269}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{b}}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (/ (- t a) (- b y))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -1e-269)
       t_1
       (if (<= t_1 0.0)
         (/ (/ (fma (- t a) z (* y x)) b) z)
         (if (<= t_1 5e+307) t_1 t_2))))))

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 = (t - a) / (b - y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -1e-269) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (fma((t - a), z, (y * x)) / b) / z;
	} else if (t_1 <= 5e+307) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -1e-269)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(Float64(t - a), z, Float64(y * x)) / b) / z);
	elseif (t_1 <= 5e+307)
		tmp = t_1;
	else
		tmp = t_2;
	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[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -1e-269], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+307], t$95$1, t$95$2]]]]]]

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 or 5e307 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 18.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999996e-270 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5e307

   1. Initial program 99.6%

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

   if -9.9999999999999996e-270 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 19.3%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

Alternative 4: 88.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{+120}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{t\_1}, \frac{\left(t - a\right) \cdot z}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot \frac{y}{b - y} - \frac{\left(t - a\right) \cdot y}{{\left(b - y\right)}^{2}}}{-z}, -1, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -2.35e+120)
     t_2
     (if (<= z 3e+68)
       (fma x (/ y t_1) (/ (* (- t a) z) t_1))
       (fma
        (/ (- (* x (/ y (- b y))) (/ (* (- t a) y) (pow (- b y) 2.0))) (- z))
        -1.0
        t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.35e+120) {
		tmp = t_2;
	} else if (z <= 3e+68) {
		tmp = fma(x, (y / t_1), (((t - a) * z) / t_1));
	} else {
		tmp = fma((((x * (y / (b - y))) - (((t - a) * y) / pow((b - y), 2.0))) / -z), -1.0, t_2);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.35e+120)
		tmp = t_2;
	elseif (z <= 3e+68)
		tmp = fma(x, Float64(y / t_1), Float64(Float64(Float64(t - a) * z) / t_1));
	else
		tmp = fma(Float64(Float64(Float64(x * Float64(y / Float64(b - y))) - Float64(Float64(Float64(t - a) * y) / (Float64(b - y) ^ 2.0))) / Float64(-z)), -1.0, t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.35e+120], t$95$2, If[LessEqual[z, 3e+68], N[(x * N[(y / t$95$1), $MachinePrecision] + N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t - a), $MachinePrecision] * y), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision] * -1.0 + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.34999999999999997e120

   1. Initial program 19.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -2.34999999999999997e120 < z < 3.0000000000000002e68

   1. Initial program 87.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 93.6%

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

   if 3.0000000000000002e68 < z

   1. Initial program 32.1%

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

   3. Taylor expanded in z around -inf

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. div-sub N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+120}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot \frac{y}{b - y} - \frac{\left(t - a\right) \cdot y}{{\left(b - y\right)}^{2}}}{-z}, -1, \frac{t - a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{+120} \lor \neg \left(z \leq 2.35 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{t\_1}, \frac{\left(t - a\right) \cdot z}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)))
   (if (or (<= z -2.35e+120) (not (<= z 2.35e+129)))
     (/ (- t a) (- b y))
     (fma x (/ y t_1) (/ (* (- t a) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double tmp;
	if ((z <= -2.35e+120) || !(z <= 2.35e+129)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(x, (y / t_1), (((t - a) * z) / t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	tmp = 0.0
	if ((z <= -2.35e+120) || !(z <= 2.35e+129))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(x, Float64(y / t_1), Float64(Float64(Float64(t - a) * z) / t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, If[Or[LessEqual[z, -2.35e+120], N[Not[LessEqual[z, 2.35e+129]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / t$95$1), $MachinePrecision] + N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.34999999999999997e120 or 2.35000000000000004e129 < z

   1. Initial program 23.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -2.34999999999999997e120 < z < 2.35000000000000004e129

   1. Initial program 84.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 91.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+120} \lor \neg \left(z \leq 2.35 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+59} \lor \neg \left(z \leq 2.15 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.95e+59) (not (<= z 2.15e-14)))
   (/ (- t a) (- b y))
   (fma x 1.0 (/ (* (- t a) z) (fma (- b y) z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.95e+59) || !(z <= 2.15e-14)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(x, 1.0, (((t - a) * z) / fma((b - y), z, y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.95e+59) || !(z <= 2.15e-14))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(x, 1.0, Float64(Float64(Float64(t - a) * z) / fma(Float64(b - y), z, y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.95e+59], N[Not[LessEqual[z, 2.15e-14]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x * 1.0 + N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95000000000000011e59 or 2.14999999999999999e-14 < z

   1. Initial program 38.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -1.95000000000000011e59 < z < 2.14999999999999999e-14

   1. Initial program 89.9%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 83.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+59} \lor \neg \left(z \leq 2.15 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(-y, z, y\right)}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -7.5e-50)
     t_1
     (if (<= z 2.3e-230)
       (/ (fma (- t a) z (* y x)) (fma (- y) z y))
       (if (<= z 2.15e-14) (fma (/ (- t a) y) z 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 <= -7.5e-50) {
		tmp = t_1;
	} else if (z <= 2.3e-230) {
		tmp = fma((t - a), z, (y * x)) / fma(-y, z, y);
	} else if (z <= 2.15e-14) {
		tmp = fma(((t - a) / y), z, 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 <= -7.5e-50)
		tmp = t_1;
	elseif (z <= 2.3e-230)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(-y), z, y));
	elseif (z <= 2.15e-14)
		tmp = fma(Float64(Float64(t - a) / y), z, x);
	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, -7.5e-50], t$95$1, If[LessEqual[z, 2.3e-230], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[((-y) * z + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.15e-14], N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.5e-50 or 2.14999999999999999e-14 < z

   1. Initial program 45.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -7.5e-50 < z < 2.2999999999999998e-230

   1. Initial program 93.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if 2.2999999999999998e-230 < z < 2.14999999999999999e-14

   1. Initial program 83.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.8%

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

Alternative 8: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{y}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -7.5e-50)
     t_1
     (if (<= z 2.3e-230)
       (/ (fma x y (* z (- t a))) y)
       (if (<= z 2.15e-14) (fma (/ (- t a) y) z 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 <= -7.5e-50) {
		tmp = t_1;
	} else if (z <= 2.3e-230) {
		tmp = fma(x, y, (z * (t - a))) / y;
	} else if (z <= 2.15e-14) {
		tmp = fma(((t - a) / y), z, 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 <= -7.5e-50)
		tmp = t_1;
	elseif (z <= 2.3e-230)
		tmp = Float64(fma(x, y, Float64(z * Float64(t - a))) / y);
	elseif (z <= 2.15e-14)
		tmp = fma(Float64(Float64(t - a) / y), z, x);
	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, -7.5e-50], t$95$1, If[LessEqual[z, 2.3e-230], N[(N[(x * y + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 2.15e-14], N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.5e-50 or 2.14999999999999999e-14 < z

   1. Initial program 45.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -7.5e-50 < z < 2.2999999999999998e-230

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 75.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 75.6

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

   7. Applied rewrites 75.6%

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

   if 2.2999999999999998e-230 < z < 2.14999999999999999e-14

   1. Initial program 83.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.8%

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

Alternative 9: 71.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-49} \lor \neg \left(z \leq 2.15 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.4e-49) (not (<= z 2.15e-14)))
   (/ (- t a) (- b y))
   (fma (/ (- t a) y) z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.4e-49) || !(z <= 2.15e-14)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(((t - a) / y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.4e-49) || !(z <= 2.15e-14))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(Float64(Float64(t - a) / y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.4e-49], N[Not[LessEqual[z, 2.15e-14]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999999e-49 or 2.14999999999999999e-14 < z

   1. Initial program 45.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -1.39999999999999999e-49 < z < 2.14999999999999999e-14

   1. Initial program 89.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-49} \lor \neg \left(z \leq 2.15 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-53} \lor \neg \left(z \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.3e-53) (not (<= z 3.4e-16)))
   (/ (- t a) (- b y))
   (fma (/ t y) z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.3e-53) || !(z <= 3.4e-16)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma((t / y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.3e-53) || !(z <= 3.4e-16))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(Float64(t / y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.3e-53], N[Not[LessEqual[z, 3.4e-16]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(t / y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000001e-53 or 3.4e-16 < z

   1. Initial program 45.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -2.3000000000000001e-53 < z < 3.4e-16

   1. Initial program 89.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 57.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-53} \lor \neg \left(z \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-53} \lor \neg \left(z \leq 3.3 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.1e-53) (not (<= z 3.3e-16)))
   (/ (- t a) (- b y))
   (fma x z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.1e-53) || !(z <= 3.3e-16)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.1e-53) || !(z <= 3.3e-16))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.1e-53], N[Not[LessEqual[z, 3.3e-16]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.09999999999999977e-53 or 3.29999999999999988e-16 < z

   1. Initial program 45.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -2.09999999999999977e-53 < z < 3.29999999999999988e-16

   1. Initial program 89.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 43.8%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-53} \lor \neg \left(z \leq 3.3 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-12} \lor \neg \left(y \leq 205\right):\\ \;\;\;\;\frac{-x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.1e-12) (not (<= y 205.0)))
   (/ (- x) (- z 1.0))
   (/ (- t a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.1e-12) || !(y <= 205.0)) {
		tmp = -x / (z - 1.0);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-2.1d-12)) .or. (.not. (y <= 205.0d0))) then
        tmp = -x / (z - 1.0d0)
    else
        tmp = (t - a) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.1e-12) || !(y <= 205.0)) {
		tmp = -x / (z - 1.0);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.1e-12) or not (y <= 205.0):
		tmp = -x / (z - 1.0)
	else:
		tmp = (t - a) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.1e-12) || !(y <= 205.0))
		tmp = Float64(Float64(-x) / Float64(z - 1.0));
	else
		tmp = Float64(Float64(t - a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.1e-12) || ~((y <= 205.0)))
		tmp = -x / (z - 1.0);
	else
		tmp = (t - a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.1e-12], N[Not[LessEqual[y, 205.0]], $MachinePrecision]], N[((-x) / N[(z - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999994e-12 or 205 < y

   1. Initial program 62.1%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 50.0

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

   5. Applied rewrites 50.0%

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

   if -2.09999999999999994e-12 < y < 205

   1. Initial program 71.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 49.4

         \[\leadsto \frac{t - a}{b} \]

   5. Applied rewrites 49.4%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-12} \lor \neg \left(y \leq 205\right):\\ \;\;\;\;\frac{-x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-53} \lor \neg \left(z \leq 2.1 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.2e-53) (not (<= z 2.1e-22))) (/ (- t a) b) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.2e-53) || !(z <= 2.1e-22)) {
		tmp = (t - a) / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-2.2d-53)) .or. (.not. (z <= 2.1d-22))) then
        tmp = (t - a) / b
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.2e-53) || !(z <= 2.1e-22)) {
		tmp = (t - a) / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -2.2e-53) or not (z <= 2.1e-22):
		tmp = (t - a) / b
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.2e-53) || !(z <= 2.1e-22))
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -2.2e-53) || ~((z <= 2.1e-22)))
		tmp = (t - a) / b;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.2e-53], N[Not[LessEqual[z, 2.1e-22]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.20000000000000018e-53 or 2.10000000000000008e-22 < z

   1. Initial program 45.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 47.4

         \[\leadsto \frac{t - a}{b} \]

   5. Applied rewrites 47.4%

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

   if -2.20000000000000018e-53 < z < 2.10000000000000008e-22

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 43.6%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-53} \lor \neg \left(z \leq 2.1 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-50} \lor \neg \left(z \leq 8.2 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.95e-50) (not (<= z 8.2e-16))) (/ t (- b y)) (fma x z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.95e-50) || !(z <= 8.2e-16)) {
		tmp = t / (b - y);
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.95e-50) || !(z <= 8.2e-16))
		tmp = Float64(t / Float64(b - y));
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.95e-50], N[Not[LessEqual[z, 8.2e-16]], $MachinePrecision]], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9500000000000001e-50 or 8.20000000000000012e-16 < z

   1. Initial program 45.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 26.7

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

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 37.9

         \[\leadsto \frac{t}{b - y} \]

   8. Applied rewrites 37.9%

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

   if -1.9500000000000001e-50 < z < 8.20000000000000012e-16

   1. Initial program 89.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 43.8%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-50} \lor \neg \left(z \leq 8.2 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-50} \lor \neg \left(z \leq 2.1 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.95e-50) (not (<= z 2.1e-22))) (/ t b) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.95e-50) || !(z <= 2.1e-22)) {
		tmp = t / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-1.95d-50)) .or. (.not. (z <= 2.1d-22))) then
        tmp = t / b
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.95e-50) || !(z <= 2.1e-22)) {
		tmp = t / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -1.95e-50) or not (z <= 2.1e-22):
		tmp = t / b
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.95e-50) || !(z <= 2.1e-22))
		tmp = Float64(t / b);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -1.95e-50) || ~((z <= 2.1e-22)))
		tmp = t / b;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.95e-50], N[Not[LessEqual[z, 2.1e-22]], $MachinePrecision]], N[(t / b), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.9500000000000001e-50 or 2.10000000000000008e-22 < z

   1. Initial program 45.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 33.0

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

   5. Applied rewrites 33.0%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 26.0

         \[\leadsto \frac{t}{b} \]

   8. Applied rewrites 26.0%

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

   if -1.9500000000000001e-50 < z < 2.10000000000000008e-22

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 43.6%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-50} \lor \neg \left(z \leq 2.1 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 25.4% accurate, 39.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 66.8%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 23.1%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 73.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z)))))

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