Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.2% → 93.1%

Time: 10.9s

Alternatives: 16

Speedup: 1.0×

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: 66.2% 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: 93.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(y, \frac{x}{t\_2}, \left(t - a\right) \cdot \frac{z}{t\_2}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+250}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma y (/ x t_2) (* (- t a) (/ z t_2)))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-229)
       t_3
       (if (<= t_3 0.0)
         t_1
         (if (<= t_3 1e+250) t_3 (if (<= t_3 INFINITY) t_4 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 t_2 = fma((b - y), z, y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma(y, (x / t_2), ((t - a) * (z / t_2)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-229) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= 1e+250) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(y, Float64(x / t_2), Float64(Float64(t - a) * Float64(z / t_2)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-229)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= 1e+250)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_4;
	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]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = 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$4 = N[(y * N[(x / t$95$2), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-229], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, 1e+250], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$1]]]]]]]]]

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 9.9999999999999992e249 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 26.5%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.5

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-229 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.9999999999999992e249

   1. Initial program 99.5%

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

   if -1.00000000000000007e-229 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 9.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 76.3%

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

Alternative 2: 90.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b - y\right)\\ t_2 := \frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{t - a}{z}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + t\_1}\\ t_4 := \mathsf{fma}\left(b - y, z, y\right)\\ t_5 := \frac{t - a}{t\_4}\\ t_6 := t\_3 \leq \infty\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, t\_5, y \cdot \frac{x}{t\_1 + y}\right)\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_6:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_6:\\ \;\;\;\;\mathsf{fma}\left(z, t\_5, y \cdot \frac{x}{t\_4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- b y)))
        (t_2
         (-
          (/ (fma (/ x z) y (- t a)) (- b y))
          (* (/ y (pow (- b y) 2.0)) (/ (- t a) z))))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y t_1)))
        (t_4 (fma (- b y) z y))
        (t_5 (/ (- t a) t_4))
        (t_6 (<= t_3 INFINITY)))
   (if (<= t_3 (- INFINITY))
     (fma z t_5 (* y (/ x (+ t_1 y))))
     (if (<= t_3 -1e-229)
       t_3
       (if (<= t_3 0.0)
         t_2
         (if t_6 t_3 (if t_6 (fma z t_5 (* y (/ x t_4))) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (b - y);
	double t_2 = (fma((x / z), y, (t - a)) / (b - y)) - ((y / pow((b - y), 2.0)) * ((t - a) / z));
	double t_3 = ((x * y) + (z * (t - a))) / (y + t_1);
	double t_4 = fma((b - y), z, y);
	double t_5 = (t - a) / t_4;
	int t_6 = t_3 <= ((double) INFINITY);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(z, t_5, (y * (x / (t_1 + y))));
	} else if (t_3 <= -1e-229) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_6) {
		tmp = t_3;
	} else if (t_6) {
		tmp = fma(z, t_5, (y * (x / t_4)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(b - y))
	t_2 = Float64(Float64(fma(Float64(x / z), y, Float64(t - a)) / Float64(b - y)) - Float64(Float64(y / (Float64(b - y) ^ 2.0)) * Float64(Float64(t - a) / z)))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + t_1))
	t_4 = fma(Float64(b - y), z, y)
	t_5 = Float64(Float64(t - a) / t_4)
	t_6 = t_3 <= Inf
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(z, t_5, Float64(y * Float64(x / Float64(t_1 + y))));
	elseif (t_3 <= -1e-229)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_6)
		tmp = t_3;
	elseif (t_6)
		tmp = fma(z, t_5, Float64(y * Float64(x / t_4)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x / z), $MachinePrecision] * y + N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t - a), $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$6 = LessEqual[t$95$3, Infinity]}, If[LessEqual[t$95$3, (-Infinity)], N[(z * t$95$5 + N[(y * N[(x / N[(t$95$1 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-229], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$2, If[t$95$6, t$95$3, If[t$95$6, N[(z * t$95$5 + N[(y * N[(x / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]]

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 18.3%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.5

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-229 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 87.8%

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

   if -1.00000000000000007e-229 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 9.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 95.2%

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

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

   1. Initial program 60.7%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 70.9

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 70.9%

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

Alternative 3: 87.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b - y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + t\_1}\\ t_3 := t\_2 \leq \infty\\ t_4 := \mathsf{fma}\left(b - y, z, y\right)\\ t_5 := \frac{t - a}{t\_4}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, t\_5, y \cdot \frac{x}{t\_1 + y}\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-x, y, \frac{\left(t - a\right) \cdot y}{b - y}\right)}{b - y} \cdot y}{z \cdot z} + \mathsf{fma}\left(\frac{y}{z}, x, t\right)}{b - y} - \frac{\frac{y \cdot \frac{t - a}{z}}{b - y} + a}{b - y}\\ \mathbf{elif}\;t\_3:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3:\\ \;\;\;\;\mathsf{fma}\left(z, t\_5, y \cdot \frac{x}{t\_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- b y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y t_1)))
        (t_3 (<= t_2 INFINITY))
        (t_4 (fma (- b y) z y))
        (t_5 (/ (- t a) t_4)))
   (if (<= t_2 (- INFINITY))
     (fma z t_5 (* y (/ x (+ t_1 y))))
     (if (<= t_2 -1e-229)
       t_2
       (if (<= t_2 0.0)
         (-
          (/
           (+
            (/
             (* (/ (fma (- x) y (/ (* (- t a) y) (- b y))) (- b y)) y)
             (* z z))
            (fma (/ y z) x t))
           (- b y))
          (/ (+ (/ (* y (/ (- t a) z)) (- b y)) a) (- b y)))
         (if t_3
           t_2
           (if t_3 (fma z t_5 (* y (/ x t_4))) (/ (- t a) (- b y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (b - y);
	double t_2 = ((x * y) + (z * (t - a))) / (y + t_1);
	int t_3 = t_2 <= ((double) INFINITY);
	double t_4 = fma((b - y), z, y);
	double t_5 = (t - a) / t_4;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(z, t_5, (y * (x / (t_1 + y))));
	} else if (t_2 <= -1e-229) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (((((fma(-x, y, (((t - a) * y) / (b - y))) / (b - y)) * y) / (z * z)) + fma((y / z), x, t)) / (b - y)) - ((((y * ((t - a) / z)) / (b - y)) + a) / (b - y));
	} else if (t_3) {
		tmp = t_2;
	} else if (t_3) {
		tmp = fma(z, t_5, (y * (x / t_4)));
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(b - y))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + t_1))
	t_3 = t_2 <= Inf
	t_4 = fma(Float64(b - y), z, y)
	t_5 = Float64(Float64(t - a) / t_4)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(z, t_5, Float64(y * Float64(x / Float64(t_1 + y))));
	elseif (t_2 <= -1e-229)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(fma(Float64(-x), y, Float64(Float64(Float64(t - a) * y) / Float64(b - y))) / Float64(b - y)) * y) / Float64(z * z)) + fma(Float64(y / z), x, t)) / Float64(b - y)) - Float64(Float64(Float64(Float64(y * Float64(Float64(t - a) / z)) / Float64(b - y)) + a) / Float64(b - y)));
	elseif (t_3)
		tmp = t_2;
	elseif (t_3)
		tmp = fma(z, t_5, Float64(y * Float64(x / t_4)));
	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[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = LessEqual[t$95$2, Infinity]}, Block[{t$95$4 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t - a), $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(z * t$95$5 + N[(y * N[(x / N[(t$95$1 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-229], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(N[(N[(N[((-x) * y + N[(N[(N[(t - a), $MachinePrecision] * y), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * x + t), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(y * N[(N[(t - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[t$95$3, t$95$2, If[t$95$3, N[(z * t$95$5 + N[(y * N[(x / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 18.3%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.5

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-229 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 87.8%

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

   if -1.00000000000000007e-229 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 33.2%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 94.6%

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

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

   1. Initial program 60.7%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 70.9

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 70.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b - y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + t\_1}\\ t_4 := t\_3 \leq \infty\\ t_5 := \mathsf{fma}\left(b - y, z, y\right)\\ t_6 := \frac{t - a}{t\_5}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, t\_6, y \cdot \frac{x}{t\_1 + y}\right)\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4:\\ \;\;\;\;\mathsf{fma}\left(z, t\_6, y \cdot \frac{x}{t\_5}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- b y)))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y t_1)))
        (t_4 (<= t_3 INFINITY))
        (t_5 (fma (- b y) z y))
        (t_6 (/ (- t a) t_5)))
   (if (<= t_3 (- INFINITY))
     (fma z t_6 (* y (/ x (+ t_1 y))))
     (if (<= t_3 -1e-229)
       t_3
       (if (<= t_3 0.0)
         t_2
         (if t_4 t_3 (if t_4 (fma z t_6 (* y (/ x t_5))) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (b - y);
	double t_2 = (t - a) / (b - y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + t_1);
	int t_4 = t_3 <= ((double) INFINITY);
	double t_5 = fma((b - y), z, y);
	double t_6 = (t - a) / t_5;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(z, t_6, (y * (x / (t_1 + y))));
	} else if (t_3 <= -1e-229) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_4) {
		tmp = t_3;
	} else if (t_4) {
		tmp = fma(z, t_6, (y * (x / t_5)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(b - y))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + t_1))
	t_4 = t_3 <= Inf
	t_5 = fma(Float64(b - y), z, y)
	t_6 = Float64(Float64(t - a) / t_5)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(z, t_6, Float64(y * Float64(x / Float64(t_1 + y))));
	elseif (t_3 <= -1e-229)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_4)
		tmp = t_3;
	elseif (t_4)
		tmp = fma(z, t_6, Float64(y * Float64(x / t_5)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = LessEqual[t$95$3, Infinity]}, Block[{t$95$5 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t - a), $MachinePrecision] / t$95$5), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(z * t$95$6 + N[(y * N[(x / N[(t$95$1 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-229], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$2, If[t$95$4, t$95$3, If[t$95$4, N[(z * t$95$6 + N[(y * N[(x / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]]

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 18.3%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.5

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-229 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 87.8%

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

   if -1.00000000000000007e-229 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 9.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 76.3%

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

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

   1. Initial program 60.7%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 70.9

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 70.9%

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

Alternative 5: 86.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := t\_3 \leq \infty\\ t_5 := \mathsf{fma}\left(z, \frac{t - a}{t\_2}, y \cdot \frac{x}{t\_2}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (<= t_3 INFINITY))
        (t_5 (fma z (/ (- t a) t_2) (* y (/ x t_2)))))
   (if (<= t_3 (- INFINITY))
     t_5
     (if (<= t_3 -1e-229)
       t_3
       (if (<= t_3 0.0) t_1 (if t_4 t_3 (if t_4 t_5 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 t_2 = fma((b - y), z, y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	int t_4 = t_3 <= ((double) INFINITY);
	double t_5 = fma(z, ((t - a) / t_2), (y * (x / t_2)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_3 <= -1e-229) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_4) {
		tmp = t_3;
	} else if (t_4) {
		tmp = t_5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = t_3 <= Inf
	t_5 = fma(z, Float64(Float64(t - a) / t_2), Float64(y * Float64(x / t_2)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_3 <= -1e-229)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_4)
		tmp = t_3;
	elseif (t_4)
		tmp = t_5;
	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]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = 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$4 = LessEqual[t$95$3, Infinity]}, Block[{t$95$5 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(y * N[(x / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$5, If[LessEqual[t$95$3, -1e-229], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$1, If[t$95$4, t$95$3, If[t$95$4, t$95$5, t$95$1]]]]]]]]]]

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

   1. Initial program 18.3%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.5

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-229 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 87.8%

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

   if -1.00000000000000007e-229 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 9.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 76.3%

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

Alternative 6: 82.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ t_4 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_5 := t\_4 \leq \infty\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{+281}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;\frac{x \cdot y + \left(z - \frac{a \cdot z}{t}\right) \cdot t}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (fma z (/ (- t a) (fma (- b y) z y)) x))
        (t_4 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_5 (<= t_4 INFINITY)))
   (if (<= t_4 -4e+281)
     t_3
     (if (<= t_4 -1e-229)
       (/ (+ (* x y) (* (- z (/ (* a z) t)) t)) t_1)
       (if (<= t_4 0.0) t_2 (if t_5 t_4 (if t_5 t_3 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = (t - a) / (b - y);
	double t_3 = fma(z, ((t - a) / fma((b - y), z, y)), x);
	double t_4 = ((x * y) + (z * (t - a))) / t_1;
	int t_5 = t_4 <= ((double) INFINITY);
	double tmp;
	if (t_4 <= -4e+281) {
		tmp = t_3;
	} else if (t_4 <= -1e-229) {
		tmp = ((x * y) + ((z - ((a * z) / t)) * t)) / t_1;
	} else if (t_4 <= 0.0) {
		tmp = t_2;
	} else if (t_5) {
		tmp = t_4;
	} else if (t_5) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), x)
	t_4 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_5 = t_4 <= Inf
	tmp = 0.0
	if (t_4 <= -4e+281)
		tmp = t_3;
	elseif (t_4 <= -1e-229)
		tmp = Float64(Float64(Float64(x * y) + Float64(Float64(z - Float64(Float64(a * z) / t)) * t)) / t_1);
	elseif (t_4 <= 0.0)
		tmp = t_2;
	elseif (t_5)
		tmp = t_4;
	elseif (t_5)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = LessEqual[t$95$4, Infinity]}, If[LessEqual[t$95$4, -4e+281], t$95$3, If[LessEqual[t$95$4, -1e-229], N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z - N[(N[(a * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.0], t$95$2, If[t$95$5, t$95$4, If[t$95$5, t$95$3, t$95$2]]]]]]]]]]

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)))) < -4.0000000000000001e281 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 96.9

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 88.4%

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

   if -4.0000000000000001e281 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-229

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 99.5%

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

   if -1.00000000000000007e-229 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 9.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 76.3%

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

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

   1. Initial program 81.2%

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

Alternative 7: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := t\_3 \leq \infty\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+281}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (fma z (/ (- t a) (fma (- b y) z y)) x))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (<= t_3 INFINITY)))
   (if (<= t_3 -4e+281)
     t_2
     (if (<= t_3 -1e-229)
       t_3
       (if (<= t_3 0.0) t_1 (if t_4 t_3 (if t_4 t_2 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 t_2 = fma(z, ((t - a) / fma((b - y), z, y)), x);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	int t_4 = t_3 <= ((double) INFINITY);
	double tmp;
	if (t_3 <= -4e+281) {
		tmp = t_2;
	} else if (t_3 <= -1e-229) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_4) {
		tmp = t_3;
	} else if (t_4) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), x)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = t_3 <= Inf
	tmp = 0.0
	if (t_3 <= -4e+281)
		tmp = t_2;
	elseif (t_3 <= -1e-229)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_4)
		tmp = t_3;
	elseif (t_4)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$3 = 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$4 = LessEqual[t$95$3, Infinity]}, If[LessEqual[t$95$3, -4e+281], t$95$2, If[LessEqual[t$95$3, -1e-229], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$1, If[t$95$4, t$95$3, If[t$95$4, t$95$2, t$95$1]]]]]]]]]

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)))) < -4.0000000000000001e281 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 96.9

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 88.4%

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

   if -4.0000000000000001e281 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-229 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 87.6%

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

   if -1.00000000000000007e-229 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 9.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 76.3%

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

Alternative 8: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{-24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-175}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{t\_1}\\ \mathbf{elif}\;z \leq 15500000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{t\_1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -1.35e-24)
     t_2
     (if (<= z -1.12e-175)
       (/ (fma t z (* y x)) t_1)
       (if (<= z 15500000000.0) (fma z (/ (- t a) t_1) x) 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 <= -1.35e-24) {
		tmp = t_2;
	} else if (z <= -1.12e-175) {
		tmp = fma(t, z, (y * x)) / t_1;
	} else if (z <= 15500000000.0) {
		tmp = fma(z, ((t - a) / t_1), x);
	} else {
		tmp = 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 <= -1.35e-24)
		tmp = t_2;
	elseif (z <= -1.12e-175)
		tmp = Float64(fma(t, z, Float64(y * x)) / t_1);
	elseif (z <= 15500000000.0)
		tmp = fma(z, Float64(Float64(t - a) / t_1), x);
	else
		tmp = 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, -1.35e-24], t$95$2, If[LessEqual[z, -1.12e-175], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 15500000000.0], N[(z * N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35000000000000003e-24 or 1.55e10 < z

   1. Initial program 38.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 80.6%

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

   if -1.35000000000000003e-24 < z < -1.1200000000000001e-175

   1. Initial program 89.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 72.0%

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

   if -1.1200000000000001e-175 < z < 1.55e10

   1. Initial program 80.0%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 83.6

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 82.3%

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

Alternative 9: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-183}:\\ \;\;\;\;\left(t - a\right) \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -3.3e+15)
     t_2
     (if (<= z -1.65e-183)
       (* (- t a) (/ z t_1))
       (if (<= z 9.2e-5) (* (/ y t_1) x) 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 <= -3.3e+15) {
		tmp = t_2;
	} else if (z <= -1.65e-183) {
		tmp = (t - a) * (z / t_1);
	} else if (z <= 9.2e-5) {
		tmp = (y / t_1) * x;
	} else {
		tmp = 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 <= -3.3e+15)
		tmp = t_2;
	elseif (z <= -1.65e-183)
		tmp = Float64(Float64(t - a) * Float64(z / t_1));
	elseif (z <= 9.2e-5)
		tmp = Float64(Float64(y / t_1) * x);
	else
		tmp = 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, -3.3e+15], t$95$2, If[LessEqual[z, -1.65e-183], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.2e-5], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3e15 or 9.20000000000000001e-5 < z

   1. Initial program 38.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 80.8%

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

   if -3.3e15 < z < -1.65e-183

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 61.5%

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

   if -1.65e-183 < z < 9.20000000000000001e-5

   1. Initial program 79.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 64.8%

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

Alternative 10: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-26} \lor \neg \left(z \leq 15500000000\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -7.2e-26) (not (<= z 15500000000.0)))
   (/ (- t a) (- b y))
   (fma z (/ (- t a) (fma (- b y) z y)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -7.2e-26) || !(z <= 15500000000.0)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(z, ((t - a) / fma((b - y), z, y)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -7.2e-26) || !(z <= 15500000000.0))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -7.2e-26], N[Not[LessEqual[z, 15500000000.0]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.2000000000000003e-26 or 1.55e10 < z

   1. Initial program 39.3%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 80.0%

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

   if -7.2000000000000003e-26 < z < 1.55e10

   1. Initial program 81.8%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 84.7

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 77.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-26} \lor \neg \left(z \leq 15500000000\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-68} \lor \neg \left(z \leq 9.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -6.8e-68) (not (<= z 9.2e-5)))
   (/ (- t a) (- b y))
   (* (/ y (fma (- b y) z y)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -6.8e-68) || !(z <= 9.2e-5)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = (y / fma((b - y), z, y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -6.8e-68) || !(z <= 9.2e-5))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -6.8e-68], N[Not[LessEqual[z, 9.2e-5]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.80000000000000037e-68 or 9.20000000000000001e-5 < z

   1. Initial program 43.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.5%

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

   if -6.80000000000000037e-68 < z < 9.20000000000000001e-5

   1. Initial program 81.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 60.9%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-68} \lor \neg \left(z \leq 9.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-137} \lor \neg \left(z \leq 2.8 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -9.5e-137) (not (<= z 2.8e-59))) (/ (- t a) (- b y)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -9.5e-137) || !(z <= 2.8e-59)) {
		tmp = (t - a) / (b - y);
	} 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 <= (-9.5d-137)) .or. (.not. (z <= 2.8d-59))) then
        tmp = (t - a) / (b - y)
    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 <= -9.5e-137) || !(z <= 2.8e-59)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -9.5e-137) or not (z <= 2.8e-59):
		tmp = (t - a) / (b - y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -9.5e-137) || !(z <= 2.8e-59))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -9.5e-137) || ~((z <= 2.8e-59)))
		tmp = (t - a) / (b - y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -9.5e-137], N[Not[LessEqual[z, 2.8e-59]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -9.5000000000000007e-137 or 2.79999999999999981e-59 < z

   1. Initial program 48.2%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 73.0%

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

   if -9.5000000000000007e-137 < z < 2.79999999999999981e-59

   1. Initial program 81.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 61.1%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-137} \lor \neg \left(z \leq 2.8 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00019 \lor \neg \left(y \leq 1.75 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -0.00019) (not (<= y 1.75e+31))) (/ x (- 1.0 z)) (/ (- t a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -0.00019) || !(y <= 1.75e+31)) {
		tmp = x / (1.0 - z);
	} 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 <= (-0.00019d0)) .or. (.not. (y <= 1.75d+31))) then
        tmp = x / (1.0d0 - z)
    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 <= -0.00019) || !(y <= 1.75e+31)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -0.00019) or not (y <= 1.75e+31):
		tmp = x / (1.0 - z)
	else:
		tmp = (t - a) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -0.00019) || !(y <= 1.75e+31))
		tmp = Float64(x / Float64(1.0 - z));
	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 <= -0.00019) || ~((y <= 1.75e+31)))
		tmp = x / (1.0 - z);
	else
		tmp = (t - a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -0.00019], N[Not[LessEqual[y, 1.75e+31]], $MachinePrecision]], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9000000000000001e-4 or 1.75e31 < y

   1. Initial program 44.8%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 56.6%

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

   if -1.9000000000000001e-4 < y < 1.75e31

   1. Initial program 75.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 58.3%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00019 \lor \neg \left(y \leq 1.75 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-7} \lor \neg \left(y \leq 4.2 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.2e-7) (not (<= y 4.2e-131))) (/ x (- 1.0 z)) (/ t b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.2e-7) || !(y <= 4.2e-131)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t / 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 <= (-3.2d-7)) .or. (.not. (y <= 4.2d-131))) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t / 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 <= -3.2e-7) || !(y <= 4.2e-131)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3.2e-7) or not (y <= 4.2e-131):
		tmp = x / (1.0 - z)
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.2e-7) || !(y <= 4.2e-131))
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3.2e-7) || ~((y <= 4.2e-131)))
		tmp = x / (1.0 - z);
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.2e-7], N[Not[LessEqual[y, 4.2e-131]], $MachinePrecision]], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(t / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2000000000000001e-7 or 4.19999999999999994e-131 < y

   1. Initial program 49.7%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 48.7%

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

   if -3.2000000000000001e-7 < y < 4.19999999999999994e-131

   1. Initial program 78.7%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 40.5%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-7} \lor \neg \left(y \leq 4.2 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-133} \lor \neg \left(z \leq 2.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -6.5e-133) (not (<= z 2.8e-9))) (/ t b) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -6.5e-133) || !(z <= 2.8e-9)) {
		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 <= (-6.5d-133)) .or. (.not. (z <= 2.8d-9))) 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 <= -6.5e-133) || !(z <= 2.8e-9)) {
		tmp = t / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -6.5e-133) or not (z <= 2.8e-9):
		tmp = t / b
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -6.5e-133) || !(z <= 2.8e-9))
		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 <= -6.5e-133) || ~((z <= 2.8e-9)))
		tmp = t / b;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -6.5e-133], N[Not[LessEqual[z, 2.8e-9]], $MachinePrecision]], N[(t / b), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -6.5000000000000002e-133 or 2.79999999999999984e-9 < z

   1. Initial program 46.7%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 29.3%

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

   if -6.5000000000000002e-133 < z < 2.79999999999999984e-9

   1. Initial program 80.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 58.2%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-133} \lor \neg \left(z \leq 2.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 25.6% 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 60.7%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 27.0%

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

Developer Target 1: 73.4% 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 2025019 
(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)))))