Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.2% → 94.0%

Time: 10.6s

Alternatives: 21

Speedup: 0.9×

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: 94.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(t - a, \frac{z}{t\_1}, y \cdot \frac{x}{t\_1}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-x, y, \frac{\left(t - a\right) \cdot y}{b - y}\right)}{b - y} \cdot y}{z \cdot z} + \mathsf{fma}\left(\frac{y}{z}, x, t\right)}{b - y} - \frac{\frac{y \cdot \frac{t - a}{z}}{b - y} + a}{b - y}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+260}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma (- t a) (/ z t_1) (* y (/ x t_1)))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -1e-277)
       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_2 5e+260)
           t_2
           (if (<= t_2 INFINITY) t_3 (/ (- t a) (- b y)))))))))

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 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma((t - a), (z / t_1), (y * (x / t_1)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -1e-277) {
		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_2 <= 5e+260) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(Float64(t - a), Float64(z / t_1), Float64(y * Float64(x / t_1)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -1e-277)
		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_2 <= 5e+260)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision] + N[(y * N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -1e-277], 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[LessEqual[t$95$2, 5e+260], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 4.9999999999999996e260 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 27.3%

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

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999969e-278 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.9999999999999996e260

   1. Initial program 99.6%

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

   if -9.99999999999999969e-278 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 11.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 88.2%

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

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

   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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.8

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

   5. Applied rewrites 84.8%

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

Alternative 2: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(t - a, \frac{z}{t\_1}, y \cdot \frac{x}{t\_1}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+260}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma (- t a) (/ z t_1) (* y (/ x t_1)))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 5e+260) t_2 (if (<= t_2 INFINITY) t_3 (/ (- t a) (- b y)))))))

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 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma((t - a), (z / t_1), (y * (x / t_1)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 5e+260) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(Float64(t - a), Float64(z / t_1), Float64(y * Float64(x / t_1)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 5e+260)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision] + N[(y * N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 5e+260], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 4.9999999999999996e260 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 27.3%

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

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.9999999999999996e260

   1. Initial program 90.4%

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

   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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.8

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

   5. Applied rewrites 84.8%

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

Alternative 3: 69.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{t\_1} \cdot y\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-147}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\left(t - a\right) \cdot \frac{z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -9000000.0)
     t_2
     (if (<= z -4.4e-33)
       (* (/ x t_1) y)
       (if (<= z -3.8e-70)
         (* (/ z (fma b z y)) (- t a))
         (if (<= z 1.15e-147)
           (/ (+ (* x y) (* z (- t a))) y)
           (if (<= z 1.35e+16) (* (- t a) (/ z t_1)) 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 <= -9000000.0) {
		tmp = t_2;
	} else if (z <= -4.4e-33) {
		tmp = (x / t_1) * y;
	} else if (z <= -3.8e-70) {
		tmp = (z / fma(b, z, y)) * (t - a);
	} else if (z <= 1.15e-147) {
		tmp = ((x * y) + (z * (t - a))) / y;
	} else if (z <= 1.35e+16) {
		tmp = (t - a) * (z / t_1);
	} 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 <= -9000000.0)
		tmp = t_2;
	elseif (z <= -4.4e-33)
		tmp = Float64(Float64(x / t_1) * y);
	elseif (z <= -3.8e-70)
		tmp = Float64(Float64(z / fma(b, z, y)) * Float64(t - a));
	elseif (z <= 1.15e-147)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / y);
	elseif (z <= 1.35e+16)
		tmp = Float64(Float64(t - a) * Float64(z / t_1));
	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, -9000000.0], t$95$2, If[LessEqual[z, -4.4e-33], N[(N[(x / t$95$1), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -3.8e-70], N[(N[(z / N[(b * z + y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e-147], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 1.35e+16], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -9e6 or 1.35e16 < z

   1. Initial program 39.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -9e6 < z < -4.40000000000000011e-33

   1. Initial program 74.7%

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

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 87.5

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. div-add-rev N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

   7. Applied rewrites 81.0%

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

   if -4.40000000000000011e-33 < z < -3.7999999999999998e-70

   1. Initial program 78.0%

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

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. div-add-rev N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. associate-*r/ N/A

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

   7. Applied rewrites 66.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 66.8%

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

   if -3.7999999999999998e-70 < z < 1.14999999999999995e-147

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 74.1%

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

   if 1.14999999999999995e-147 < z < 1.35e16

   1. Initial program 89.9%

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

   3. Taylor expanded in 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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 67.9

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

   5. Applied rewrites 67.9%

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

Alternative 4: 69.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{z}{\mathsf{fma}\left(b, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{if}\;z \leq -9000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b - y, z, y\right)} \cdot y\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-70}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-147}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))) (t_2 (* (/ z (fma b z y)) (- t a))))
   (if (<= z -9000000.0)
     t_1
     (if (<= z -4.4e-33)
       (* (/ x (fma (- b y) z y)) y)
       (if (<= z -3.8e-70)
         t_2
         (if (<= z 1.15e-147)
           (/ (+ (* x y) (* z (- t a))) y)
           (if (<= z 1.0) 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 = (z / fma(b, z, y)) * (t - a);
	double tmp;
	if (z <= -9000000.0) {
		tmp = t_1;
	} else if (z <= -4.4e-33) {
		tmp = (x / fma((b - y), z, y)) * y;
	} else if (z <= -3.8e-70) {
		tmp = t_2;
	} else if (z <= 1.15e-147) {
		tmp = ((x * y) + (z * (t - a))) / y;
	} else if (z <= 1.0) {
		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 = Float64(Float64(z / fma(b, z, y)) * Float64(t - a))
	tmp = 0.0
	if (z <= -9000000.0)
		tmp = t_1;
	elseif (z <= -4.4e-33)
		tmp = Float64(Float64(x / fma(Float64(b - y), z, y)) * y);
	elseif (z <= -3.8e-70)
		tmp = t_2;
	elseif (z <= 1.15e-147)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / y);
	elseif (z <= 1.0)
		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[(N[(z / N[(b * z + y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9000000.0], t$95$1, If[LessEqual[z, -4.4e-33], N[(N[(x / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -3.8e-70], t$95$2, If[LessEqual[z, 1.15e-147], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 1.0], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9e6 or 1 < z

   1. Initial program 40.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -9e6 < z < -4.40000000000000011e-33

   1. Initial program 74.7%

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

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 87.5

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. div-add-rev N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

   7. Applied rewrites 81.0%

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

   if -4.40000000000000011e-33 < z < -3.7999999999999998e-70 or 1.14999999999999995e-147 < z < 1

   1. Initial program 87.5%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. div-add-rev N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. associate-*r/ N/A

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

   7. Applied rewrites 68.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 68.2%

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

   if -3.7999999999999998e-70 < z < 1.14999999999999995e-147

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 74.1%

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

Alternative 5: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{z}{\mathsf{fma}\left(b, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{if}\;z \leq -9000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{t\_1} \cdot y\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-175}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (* (/ z (fma b z y)) (- t a))))
   (if (<= z -9000000.0)
     t_2
     (if (<= z -4.4e-33)
       (* (/ x t_1) y)
       (if (<= z -4.5e-175)
         t_3
         (if (<= z 6e-197) (* (/ y t_1) x) (if (<= z 1.0) t_3 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 t_3 = (z / fma(b, z, y)) * (t - a);
	double tmp;
	if (z <= -9000000.0) {
		tmp = t_2;
	} else if (z <= -4.4e-33) {
		tmp = (x / t_1) * y;
	} else if (z <= -4.5e-175) {
		tmp = t_3;
	} else if (z <= 6e-197) {
		tmp = (y / t_1) * x;
	} else if (z <= 1.0) {
		tmp = t_3;
	} 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))
	t_3 = Float64(Float64(z / fma(b, z, y)) * Float64(t - a))
	tmp = 0.0
	if (z <= -9000000.0)
		tmp = t_2;
	elseif (z <= -4.4e-33)
		tmp = Float64(Float64(x / t_1) * y);
	elseif (z <= -4.5e-175)
		tmp = t_3;
	elseif (z <= 6e-197)
		tmp = Float64(Float64(y / t_1) * x);
	elseif (z <= 1.0)
		tmp = t_3;
	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]}, Block[{t$95$3 = N[(N[(z / N[(b * z + y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9000000.0], t$95$2, If[LessEqual[z, -4.4e-33], N[(N[(x / t$95$1), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -4.5e-175], t$95$3, If[LessEqual[z, 6e-197], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.0], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9e6 or 1 < z

   1. Initial program 40.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -9e6 < z < -4.40000000000000011e-33

   1. Initial program 74.7%

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

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 87.5

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. div-add-rev N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

   7. Applied rewrites 81.0%

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

   if -4.40000000000000011e-33 < z < -4.49999999999999998e-175 or 6.00000000000000051e-197 < z < 1

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. div-add-rev N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. associate-*r/ N/A

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 68.4%

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

   if -4.49999999999999998e-175 < z < 6.00000000000000051e-197

   1. Initial program 89.1%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

Alternative 6: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+40} \lor \neg \left(z \leq 5 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.4e+40) (not (<= z 5e+19)))
   (/ (- t a) (- b y))
   (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.4e+40) || !(z <= 5e+19)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	}
	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 <= (-4.4d+40)) .or. (.not. (z <= 5d+19))) then
        tmp = (t - a) / (b - y)
    else
        tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
    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 <= -4.4e+40) || !(z <= 5e+19)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -4.4e+40) or not (z <= 5e+19):
		tmp = (t - a) / (b - y)
	else:
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -4.4e+40) || !(z <= 5e+19))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -4.4e+40) || ~((z <= 5e+19)))
		tmp = (t - a) / (b - y);
	else
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -4.4e+40], N[Not[LessEqual[z, 5e+19]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if z < -4.3999999999999998e40 or 5e19 < z

   1. Initial program 37.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if -4.3999999999999998e40 < z < 5e19

   1. Initial program 87.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+40} \lor \neg \left(z \leq 5 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b - y, z, y\right)} \cdot y\\ \mathbf{elif}\;z \leq -1.52 \cdot 10^{-276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{y}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, z, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -9000000.0)
     t_1
     (if (<= z -3.4e-70)
       (* (/ x (fma (- b y) z y)) y)
       (if (<= z -1.52e-276)
         (/ (fma x y (* z t)) y)
         (if (<= z 6.2e-34) (/ (fma (- a) z (* x y)) y) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -9000000.0) {
		tmp = t_1;
	} else if (z <= -3.4e-70) {
		tmp = (x / fma((b - y), z, y)) * y;
	} else if (z <= -1.52e-276) {
		tmp = fma(x, y, (z * t)) / y;
	} else if (z <= 6.2e-34) {
		tmp = fma(-a, z, (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -9000000.0)
		tmp = t_1;
	elseif (z <= -3.4e-70)
		tmp = Float64(Float64(x / fma(Float64(b - y), z, y)) * y);
	elseif (z <= -1.52e-276)
		tmp = Float64(fma(x, y, Float64(z * t)) / y);
	elseif (z <= 6.2e-34)
		tmp = Float64(fma(Float64(-a), z, Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9000000.0], t$95$1, If[LessEqual[z, -3.4e-70], N[(N[(x / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -1.52e-276], N[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 6.2e-34], N[(N[((-a) * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9e6 or 6.1999999999999996e-34 < z

   1. Initial program 42.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if -9e6 < z < -3.39999999999999995e-70

   1. Initial program 76.5%

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

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. div-add-rev N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

   7. Applied rewrites 54.7%

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

   if -3.39999999999999995e-70 < z < -1.51999999999999994e-276

   1. Initial program 96.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.9

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

   8. Applied rewrites 64.9%

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

   if -1.51999999999999994e-276 < z < 6.1999999999999996e-34

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 59.9

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

   8. Applied rewrites 59.9%

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

Alternative 8: 64.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -135000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-z, y, y\right)} \cdot x\\ \mathbf{elif}\;z \leq -1.52 \cdot 10^{-276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{y}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, z, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -135000.0)
     t_1
     (if (<= z -3.4e-70)
       (* (/ y (fma (- z) y y)) x)
       (if (<= z -1.52e-276)
         (/ (fma x y (* z t)) y)
         (if (<= z 6.2e-34) (/ (fma (- a) z (* x y)) y) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -135000.0) {
		tmp = t_1;
	} else if (z <= -3.4e-70) {
		tmp = (y / fma(-z, y, y)) * x;
	} else if (z <= -1.52e-276) {
		tmp = fma(x, y, (z * t)) / y;
	} else if (z <= 6.2e-34) {
		tmp = fma(-a, z, (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -135000.0)
		tmp = t_1;
	elseif (z <= -3.4e-70)
		tmp = Float64(Float64(y / fma(Float64(-z), y, y)) * x);
	elseif (z <= -1.52e-276)
		tmp = Float64(fma(x, y, Float64(z * t)) / y);
	elseif (z <= 6.2e-34)
		tmp = Float64(fma(Float64(-a), z, Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -135000.0], t$95$1, If[LessEqual[z, -3.4e-70], N[(N[(y / N[((-z) * y + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, -1.52e-276], N[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 6.2e-34], N[(N[((-a) * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -135000 or 6.1999999999999996e-34 < z

   1. Initial program 43.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -135000 < z < -3.39999999999999995e-70

   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 x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 54.0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{y}{\left(\mathsf{neg}\left(y \cdot z\right)\right) + y} \cdot x \]

      3. *-commutative N/A

         \[\leadsto \frac{y}{\left(\mathsf{neg}\left(z \cdot y\right)\right) + y} \cdot x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{y}{\left(\mathsf{neg}\left(z\right)\right) \cdot y + y} \cdot x \]

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 47.8

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

   8. Applied rewrites 47.8%

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

   if -3.39999999999999995e-70 < z < -1.51999999999999994e-276

   1. Initial program 96.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.9

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

   8. Applied rewrites 64.9%

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

   if -1.51999999999999994e-276 < z < 6.1999999999999996e-34

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 59.9

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

   8. Applied rewrites 59.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -135000:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-z, y, y\right)} \cdot x\\ \mathbf{elif}\;z \leq -1.52 \cdot 10^{-276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{y}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, z, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.046:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x \cdot \frac{y}{y}\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -0.046)
     t_1
     (if (<= z 7.5e-125)
       (fma z (/ (- t a) y) (* x (/ y y)))
       (if (<= z 1.35e+16) (* (- t a) (/ z (fma (- b y) z y))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -0.046) {
		tmp = t_1;
	} else if (z <= 7.5e-125) {
		tmp = fma(z, ((t - a) / y), (x * (y / y)));
	} else if (z <= 1.35e+16) {
		tmp = (t - a) * (z / fma((b - y), z, y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -0.046)
		tmp = t_1;
	elseif (z <= 7.5e-125)
		tmp = fma(z, Float64(Float64(t - a) / y), Float64(x * Float64(y / y)));
	elseif (z <= 1.35e+16)
		tmp = Float64(Float64(t - a) * Float64(z / fma(Float64(b - y), z, y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.046], t$95$1, If[LessEqual[z, 7.5e-125], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + N[(x * N[(y / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.35e+16], N[(N[(t - a), $MachinePrecision] * N[(z / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.045999999999999999 or 1.35e16 < z

   1. Initial program 41.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -0.045999999999999999 < z < 7.5e-125

   1. Initial program 85.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 61.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 70.9

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

   7. Applied rewrites 70.9%

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

   if 7.5e-125 < z < 1.35e16

   1. Initial program 93.6%

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

   3. Taylor expanded in 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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 70.2

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

   5. Applied rewrites 70.2%

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

Alternative 10: 70.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.9 \cdot 10^{+31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-197}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{t\_1}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\left(t - a\right) \cdot \frac{z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -3.9e+31)
     t_2
     (if (<= z 6.8e-197)
       (/ (fma t z (* y x)) t_1)
       (if (<= z 1.35e+16) (* (- t a) (/ z t_1)) 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 <= -3.9e+31) {
		tmp = t_2;
	} else if (z <= 6.8e-197) {
		tmp = fma(t, z, (y * x)) / t_1;
	} else if (z <= 1.35e+16) {
		tmp = (t - a) * (z / t_1);
	} 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 <= -3.9e+31)
		tmp = t_2;
	elseif (z <= 6.8e-197)
		tmp = Float64(fma(t, z, Float64(y * x)) / t_1);
	elseif (z <= 1.35e+16)
		tmp = Float64(Float64(t - a) * Float64(z / t_1));
	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, -3.9e+31], t$95$2, If[LessEqual[z, 6.8e-197], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 1.35e+16], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.89999999999999999e31 or 1.35e16 < z

   1. Initial program 38.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if -3.89999999999999999e31 < z < 6.7999999999999996e-197

   1. Initial program 86.6%

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

   3. Taylor expanded in 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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if 6.7999999999999996e-197 < z < 1.35e16

   1. Initial program 90.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 67.8

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

   5. Applied rewrites 67.8%

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

Alternative 11: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.15:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-255}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, z, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -0.15)
     t_1
     (if (<= z 1.65e-255)
       (* (/ y (fma (- b y) z y)) x)
       (if (<= z 6.2e-34) (/ (fma (- a) z (* x y)) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -0.15) {
		tmp = t_1;
	} else if (z <= 1.65e-255) {
		tmp = (y / fma((b - y), z, y)) * x;
	} else if (z <= 6.2e-34) {
		tmp = fma(-a, z, (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -0.15)
		tmp = t_1;
	elseif (z <= 1.65e-255)
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	elseif (z <= 6.2e-34)
		tmp = Float64(fma(Float64(-a), z, Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.15], t$95$1, If[LessEqual[z, 1.65e-255], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 6.2e-34], N[(N[((-a) * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.149999999999999994 or 6.1999999999999996e-34 < z

   1. Initial program 43.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -0.149999999999999994 < z < 1.64999999999999994e-255

   1. Initial program 84.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if 1.64999999999999994e-255 < z < 6.1999999999999996e-34

   1. Initial program 91.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 55.9

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

   8. Applied rewrites 55.9%

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

Alternative 12: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -135000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-123}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-z, y, y\right)} \cdot x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -135000.0)
     t_1
     (if (<= z 3.9e-123)
       (* (/ y (fma (- z) y y)) x)
       (if (<= z 2.1e-29) (/ (* (- t a) z) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -135000.0) {
		tmp = t_1;
	} else if (z <= 3.9e-123) {
		tmp = (y / fma(-z, y, y)) * x;
	} else if (z <= 2.1e-29) {
		tmp = ((t - a) * z) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -135000.0)
		tmp = t_1;
	elseif (z <= 3.9e-123)
		tmp = Float64(Float64(y / fma(Float64(-z), y, y)) * x);
	elseif (z <= 2.1e-29)
		tmp = Float64(Float64(Float64(t - a) * z) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -135000.0], t$95$1, If[LessEqual[z, 3.9e-123], N[(N[(y / N[((-z) * y + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 2.1e-29], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -135000 or 2.09999999999999989e-29 < z

   1. Initial program 42.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -135000 < z < 3.89999999999999976e-123

   1. Initial program 85.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{y}{\left(\mathsf{neg}\left(y \cdot z\right)\right) + y} \cdot x \]

      3. *-commutative N/A

         \[\leadsto \frac{y}{\left(\mathsf{neg}\left(z \cdot y\right)\right) + y} \cdot x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{y}{\left(\mathsf{neg}\left(z\right)\right) \cdot y + y} \cdot x \]

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 52.2

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

   8. Applied rewrites 52.2%

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

   if 3.89999999999999976e-123 < z < 2.09999999999999989e-29

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   8. Applied rewrites 50.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -135000:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-123}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-z, y, y\right)} \cdot x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -135000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -135000.0)
     t_1
     (if (<= z 3.9e-123)
       (/ x (- 1.0 z))
       (if (<= z 2.1e-29) (/ (* (- t a) z) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -135000.0) {
		tmp = t_1;
	} else if (z <= 3.9e-123) {
		tmp = x / (1.0 - z);
	} else if (z <= 2.1e-29) {
		tmp = ((t - a) * z) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-135000.0d0)) then
        tmp = t_1
    else if (z <= 3.9d-123) then
        tmp = x / (1.0d0 - z)
    else if (z <= 2.1d-29) then
        tmp = ((t - a) * z) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -135000.0) {
		tmp = t_1;
	} else if (z <= 3.9e-123) {
		tmp = x / (1.0 - z);
	} else if (z <= 2.1e-29) {
		tmp = ((t - a) * z) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -135000.0:
		tmp = t_1
	elif z <= 3.9e-123:
		tmp = x / (1.0 - z)
	elif z <= 2.1e-29:
		tmp = ((t - a) * z) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -135000.0)
		tmp = t_1;
	elseif (z <= 3.9e-123)
		tmp = Float64(x / Float64(1.0 - z));
	elseif (z <= 2.1e-29)
		tmp = Float64(Float64(Float64(t - a) * z) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -135000.0)
		tmp = t_1;
	elseif (z <= 3.9e-123)
		tmp = x / (1.0 - z);
	elseif (z <= 2.1e-29)
		tmp = ((t - a) * z) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -135000.0], t$95$1, If[LessEqual[z, 3.9e-123], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e-29], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -135000 or 2.09999999999999989e-29 < z

   1. Initial program 42.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -135000 < z < 3.89999999999999976e-123

   1. Initial program 85.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if 3.89999999999999976e-123 < z < 2.09999999999999989e-29

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   8. Applied rewrites 50.3%

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

Alternative 14: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -135000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{z}{y} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -135000.0)
     t_1
     (if (<= z 1.3e-122)
       (/ x (- 1.0 z))
       (if (<= z 2.1e-29) (* (/ z y) (- t a)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -135000.0) {
		tmp = t_1;
	} else if (z <= 1.3e-122) {
		tmp = x / (1.0 - z);
	} else if (z <= 2.1e-29) {
		tmp = (z / y) * (t - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-135000.0d0)) then
        tmp = t_1
    else if (z <= 1.3d-122) then
        tmp = x / (1.0d0 - z)
    else if (z <= 2.1d-29) then
        tmp = (z / y) * (t - a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -135000.0) {
		tmp = t_1;
	} else if (z <= 1.3e-122) {
		tmp = x / (1.0 - z);
	} else if (z <= 2.1e-29) {
		tmp = (z / y) * (t - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -135000.0:
		tmp = t_1
	elif z <= 1.3e-122:
		tmp = x / (1.0 - z)
	elif z <= 2.1e-29:
		tmp = (z / y) * (t - a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -135000.0)
		tmp = t_1;
	elseif (z <= 1.3e-122)
		tmp = Float64(x / Float64(1.0 - z));
	elseif (z <= 2.1e-29)
		tmp = Float64(Float64(z / y) * Float64(t - a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -135000.0)
		tmp = t_1;
	elseif (z <= 1.3e-122)
		tmp = x / (1.0 - z);
	elseif (z <= 2.1e-29)
		tmp = (z / y) * (t - a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -135000.0], t$95$1, If[LessEqual[z, 1.3e-122], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e-29], N[(N[(z / y), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -135000 or 2.09999999999999989e-29 < z

   1. Initial program 42.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -135000 < z < 1.29999999999999988e-122

   1. Initial program 85.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if 1.29999999999999988e-122 < z < 2.09999999999999989e-29

   1. Initial program 95.6%

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

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 85.2

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. div-add-rev N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. associate-*r/ N/A

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative 48.2

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

      2. *-commutative 48.2

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

   10. Applied rewrites 48.2%

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

Alternative 15: 54.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t - a}{-y}\\ \mathbf{elif}\;y \leq 24000000:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -2.05e+96)
     t_1
     (if (<= y -4e-42)
       (/ (- t a) (- y))
       (if (<= y 24000000.0) (/ (- t a) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -2.05e+96) {
		tmp = t_1;
	} else if (y <= -4e-42) {
		tmp = (t - a) / -y;
	} else if (y <= 24000000.0) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-2.05d+96)) then
        tmp = t_1
    else if (y <= (-4d-42)) then
        tmp = (t - a) / -y
    else if (y <= 24000000.0d0) then
        tmp = (t - a) / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -2.05e+96) {
		tmp = t_1;
	} else if (y <= -4e-42) {
		tmp = (t - a) / -y;
	} else if (y <= 24000000.0) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -2.05e+96:
		tmp = t_1
	elif y <= -4e-42:
		tmp = (t - a) / -y
	elif y <= 24000000.0:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -2.05e+96)
		tmp = t_1;
	elseif (y <= -4e-42)
		tmp = Float64(Float64(t - a) / Float64(-y));
	elseif (y <= 24000000.0)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -2.05e+96)
		tmp = t_1;
	elseif (y <= -4e-42)
		tmp = (t - a) / -y;
	elseif (y <= 24000000.0)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.05e+96], t$95$1, If[LessEqual[y, -4e-42], N[(N[(t - a), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[y, 24000000.0], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.04999999999999999e96 or 2.4e7 < y

   1. Initial program 51.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 52.0

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

   5. Applied rewrites 52.0%

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

   if -2.04999999999999999e96 < y < -4.00000000000000015e-42

   1. Initial program 53.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{t - a}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 41.9

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

   8. Applied rewrites 41.9%

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

   if -4.00000000000000015e-42 < y < 2.4e7

   1. Initial program 77.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 62.5

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

   5. Applied rewrites 62.5%

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

Alternative 16: 41.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+55}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= t -1.2e-19)
     t_1
     (if (<= t 8e+15) (/ x (- 1.0 z)) (if (<= t 2.9e+55) (/ (- a) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (t <= -1.2e-19) {
		tmp = t_1;
	} else if (t <= 8e+15) {
		tmp = x / (1.0 - z);
	} else if (t <= 2.9e+55) {
		tmp = -a / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (t <= (-1.2d-19)) then
        tmp = t_1
    else if (t <= 8d+15) then
        tmp = x / (1.0d0 - z)
    else if (t <= 2.9d+55) then
        tmp = -a / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (t <= -1.2e-19) {
		tmp = t_1;
	} else if (t <= 8e+15) {
		tmp = x / (1.0 - z);
	} else if (t <= 2.9e+55) {
		tmp = -a / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if t <= -1.2e-19:
		tmp = t_1
	elif t <= 8e+15:
		tmp = x / (1.0 - z)
	elif t <= 2.9e+55:
		tmp = -a / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (t <= -1.2e-19)
		tmp = t_1;
	elseif (t <= 8e+15)
		tmp = Float64(x / Float64(1.0 - z));
	elseif (t <= 2.9e+55)
		tmp = Float64(Float64(-a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (t <= -1.2e-19)
		tmp = t_1;
	elseif (t <= 8e+15)
		tmp = x / (1.0 - z);
	elseif (t <= 2.9e+55)
		tmp = -a / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.2e-19], t$95$1, If[LessEqual[t, 8e+15], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.9e+55], N[((-a) / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.20000000000000011e-19 or 2.8999999999999999e55 < t

   1. Initial program 60.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 52.5%

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

   if -1.20000000000000011e-19 < t < 8e15

   1. Initial program 68.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 43.5

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

   5. Applied rewrites 43.5%

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

   if 8e15 < t < 2.8999999999999999e55

   1. Initial program 56.7%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

   6. Taylor expanded in a around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{a}{b}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot a}{b} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a}{b} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{b} \]

      4. lower-neg.f64 73.3

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

   8. Applied rewrites 73.3%

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

Alternative 17: 42.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -6.5e-44)
     t_1
     (if (<= y 2.8e-171) (/ t b) (if (<= y 1.6e-102) (/ (- a) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -6.5e-44) {
		tmp = t_1;
	} else if (y <= 2.8e-171) {
		tmp = t / b;
	} else if (y <= 1.6e-102) {
		tmp = -a / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-6.5d-44)) then
        tmp = t_1
    else if (y <= 2.8d-171) then
        tmp = t / b
    else if (y <= 1.6d-102) then
        tmp = -a / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -6.5e-44) {
		tmp = t_1;
	} else if (y <= 2.8e-171) {
		tmp = t / b;
	} else if (y <= 1.6e-102) {
		tmp = -a / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -6.5e-44:
		tmp = t_1
	elif y <= 2.8e-171:
		tmp = t / b
	elif y <= 1.6e-102:
		tmp = -a / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -6.5e-44)
		tmp = t_1;
	elseif (y <= 2.8e-171)
		tmp = Float64(t / b);
	elseif (y <= 1.6e-102)
		tmp = Float64(Float64(-a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -6.5e-44)
		tmp = t_1;
	elseif (y <= 2.8e-171)
		tmp = t / b;
	elseif (y <= 1.6e-102)
		tmp = -a / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-44], t$95$1, If[LessEqual[y, 2.8e-171], N[(t / b), $MachinePrecision], If[LessEqual[y, 1.6e-102], N[((-a) / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5e-44 or 1.59999999999999993e-102 < y

   1. Initial program 56.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 43.5

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

   5. Applied rewrites 43.5%

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

   if -6.5e-44 < y < 2.80000000000000023e-171

   1. Initial program 74.7%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 42.0

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

   8. Applied rewrites 42.0%

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

   if 2.80000000000000023e-171 < y < 1.59999999999999993e-102

   1. Initial program 77.7%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in a around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{a}{b}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot a}{b} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a}{b} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{b} \]

      4. lower-neg.f64 43.0

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

   8. Applied rewrites 43.0%

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

Alternative 18: 64.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -135000 \lor \neg \left(z \leq 1.8 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -135000.0) (not (<= z 1.8e-131)))
   (/ (- t a) (- b y))
   (/ x (- 1.0 z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -135000.0) || !(z <= 1.8e-131)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x / (1.0 - z);
	}
	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 <= (-135000.0d0)) .or. (.not. (z <= 1.8d-131))) then
        tmp = (t - a) / (b - y)
    else
        tmp = x / (1.0d0 - z)
    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 <= -135000.0) || !(z <= 1.8e-131)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x / (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -135000.0) or not (z <= 1.8e-131):
		tmp = (t - a) / (b - y)
	else:
		tmp = x / (1.0 - z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -135000.0) || !(z <= 1.8e-131))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(x / Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -135000.0) || ~((z <= 1.8e-131)))
		tmp = (t - a) / (b - y);
	else
		tmp = x / (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -135000.0], N[Not[LessEqual[z, 1.8e-131]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -135000 or 1.8e-131 < z

   1. Initial program 51.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -135000 < z < 1.8e-131

   1. Initial program 85.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 52.7

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

   5. Applied rewrites 52.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -135000 \lor \neg \left(z \leq 1.8 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-42} \lor \neg \left(y \leq 24000000\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4e-42) (not (<= y 24000000.0))) (/ 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 <= -4e-42) || !(y <= 24000000.0)) {
		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 <= (-4d-42)) .or. (.not. (y <= 24000000.0d0))) 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 <= -4e-42) || !(y <= 24000000.0)) {
		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 <= -4e-42) or not (y <= 24000000.0):
		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 <= -4e-42) || !(y <= 24000000.0))
		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 <= -4e-42) || ~((y <= 24000000.0)))
		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, -4e-42], N[Not[LessEqual[y, 24000000.0]], $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 < -4.00000000000000015e-42 or 2.4e7 < y

   1. Initial program 51.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 46.4

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

   5. Applied rewrites 46.4%

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

   if -4.00000000000000015e-42 < y < 2.4e7

   1. Initial program 77.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 62.5

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

   5. Applied rewrites 62.5%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-42} \lor \neg \left(y \leq 24000000\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -320000000 \lor \neg \left(z \leq 7.5 \cdot 10^{-125}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -320000000.0) (not (<= z 7.5e-125))) (/ t b) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -320000000.0) || !(z <= 7.5e-125)) {
		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 <= (-320000000.0d0)) .or. (.not. (z <= 7.5d-125))) 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 <= -320000000.0) || !(z <= 7.5e-125)) {
		tmp = t / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -320000000.0) or not (z <= 7.5e-125):
		tmp = t / b
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -320000000.0) || !(z <= 7.5e-125))
		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 <= -320000000.0) || ~((z <= 7.5e-125)))
		tmp = t / b;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -320000000.0], N[Not[LessEqual[z, 7.5e-125]], $MachinePrecision]], N[(t / b), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -3.2e8 or 7.5e-125 < z

   1. Initial program 49.8%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 34.4

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

   5. Applied rewrites 34.4%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 29.1

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

   8. Applied rewrites 29.1%

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

   if -3.2e8 < z < 7.5e-125

   1. Initial program 86.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 48.8%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -320000000 \lor \neg \left(z \leq 7.5 \cdot 10^{-125}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 25.3% 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 64.1%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 22.6%

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

Developer Target 1: 73.9% 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 2025026 
(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)))))