Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 67.4% → 94.0%

Time: 7.7s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \mathsf{fma}\left(\frac{t - a}{t\_1}, z, \frac{x}{t\_1} \cdot y\right)\\ t_3 := z \cdot \left(t - a\right)\\ t_4 := \frac{x \cdot y + t\_3}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{+280}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t\_3\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1 \cdot \frac{x \cdot y}{b - y} - -1 \cdot \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z}, \frac{t}{b - y}\right) - \frac{a}{b - y}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (fma (/ (- t a) t_1) z (* (/ x t_1) y)))
        (t_3 (* z (- t a)))
        (t_4 (/ (+ (* x y) t_3) (+ y (* z (- b y))))))
   (if (<= t_4 -4e+280)
     t_2
     (if (<= t_4 -4e-299)
       (/ (fma x y t_3) (fma z (- b y) y))
       (if (<= t_4 0.0)
         (-
          (fma
           -1.0
           (/
            (-
             (* -1.0 (/ (* x y) (- b y)))
             (* -1.0 (/ (* y (- t a)) (pow (- b y) 2.0))))
            z)
           (/ t (- b y)))
          (/ a (- b y)))
         (if (<= t_4 5e+282)
           (/ (fma (- t a) z (* y x)) t_1)
           (if (<= t_4 INFINITY) t_2 (/ (- 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 = fma(((t - a) / t_1), z, ((x / t_1) * y));
	double t_3 = z * (t - a);
	double t_4 = ((x * y) + t_3) / (y + (z * (b - y)));
	double tmp;
	if (t_4 <= -4e+280) {
		tmp = t_2;
	} else if (t_4 <= -4e-299) {
		tmp = fma(x, y, t_3) / fma(z, (b - y), y);
	} else if (t_4 <= 0.0) {
		tmp = fma(-1.0, (((-1.0 * ((x * y) / (b - y))) - (-1.0 * ((y * (t - a)) / pow((b - y), 2.0)))) / z), (t / (b - y))) - (a / (b - y));
	} else if (t_4 <= 5e+282) {
		tmp = fma((t - a), z, (y * x)) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} 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 = fma(Float64(Float64(t - a) / t_1), z, Float64(Float64(x / t_1) * y))
	t_3 = Float64(z * Float64(t - a))
	t_4 = Float64(Float64(Float64(x * y) + t_3) / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if (t_4 <= -4e+280)
		tmp = t_2;
	elseif (t_4 <= -4e-299)
		tmp = Float64(fma(x, y, t_3) / fma(z, Float64(b - y), y));
	elseif (t_4 <= 0.0)
		tmp = Float64(fma(-1.0, Float64(Float64(Float64(-1.0 * Float64(Float64(x * y) / Float64(b - y))) - Float64(-1.0 * Float64(Float64(y * Float64(t - a)) / (Float64(b - y) ^ 2.0)))) / z), Float64(t / Float64(b - y))) - Float64(a / Float64(b - y)));
	elseif (t_4 <= 5e+282)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / t_1);
	elseif (t_4 <= Inf)
		tmp = t_2;
	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[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] * z + N[(N[(x / t$95$1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * y), $MachinePrecision] + t$95$3), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e+280], t$95$2, If[LessEqual[t$95$4, -4e-299], N[(N[(x * y + t$95$3), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(x * y), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(y * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+282], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.0000000000000001e280 or 4.99999999999999978e282 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

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

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

      17. distribute-rgt-neg-in N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

   if -4.0000000000000001e280 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -3.99999999999999997e-299

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

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

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

      17. distribute-rgt-neg-in N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

   5. Applied rewrites 67.4%

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

   if -3.99999999999999997e-299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 67.4%

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

   2. Taylor expanded in z around -inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 47.4%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.99999999999999978e282

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. sub-flip N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 67.4

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

   3. Applied rewrites 67.4%

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

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

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

Alternative 2: 93.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \mathsf{fma}\left(\frac{t - a}{t\_2}, z, \frac{x}{t\_2} \cdot y\right)\\ t_4 := z \cdot \left(t - a\right)\\ t_5 := \frac{x \cdot y + t\_4}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t\_5 \leq -4 \cdot 10^{+280}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t\_4\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{t\_2}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (fma (- b y) z y))
        (t_3 (fma (/ (- t a) t_2) z (* (/ x t_2) y)))
        (t_4 (* z (- t a)))
        (t_5 (/ (+ (* x y) t_4) (+ y (* z (- b y))))))
   (if (<= t_5 -4e+280)
     t_3
     (if (<= t_5 -4e-299)
       (/ (fma x y t_4) (fma z (- b y) y))
       (if (<= t_5 0.0)
         t_1
         (if (<= t_5 5e+282)
           (/ (fma (- t a) z (* y x)) t_2)
           (if (<= t_5 INFINITY) t_3 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = fma((b - y), z, y);
	double t_3 = fma(((t - a) / t_2), z, ((x / t_2) * y));
	double t_4 = z * (t - a);
	double t_5 = ((x * y) + t_4) / (y + (z * (b - y)));
	double tmp;
	if (t_5 <= -4e+280) {
		tmp = t_3;
	} else if (t_5 <= -4e-299) {
		tmp = fma(x, y, t_4) / fma(z, (b - y), y);
	} else if (t_5 <= 0.0) {
		tmp = t_1;
	} else if (t_5 <= 5e+282) {
		tmp = fma((t - a), z, (y * x)) / t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = fma(Float64(Float64(t - a) / t_2), z, Float64(Float64(x / t_2) * y))
	t_4 = Float64(z * Float64(t - a))
	t_5 = Float64(Float64(Float64(x * y) + t_4) / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if (t_5 <= -4e+280)
		tmp = t_3;
	elseif (t_5 <= -4e-299)
		tmp = Float64(fma(x, y, t_4) / fma(z, Float64(b - y), y));
	elseif (t_5 <= 0.0)
		tmp = t_1;
	elseif (t_5 <= 5e+282)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / t_2);
	elseif (t_5 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(N[(x / t$95$2), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(x * y), $MachinePrecision] + t$95$4), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -4e+280], t$95$3, If[LessEqual[t$95$5, -4e-299], N[(N[(x * y + t$95$4), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], t$95$1, If[LessEqual[t$95$5, 5e+282], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$3, t$95$1]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.0000000000000001e280 or 4.99999999999999978e282 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

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

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

      17. distribute-rgt-neg-in N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

   if -4.0000000000000001e280 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -3.99999999999999997e-299

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

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

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

      17. distribute-rgt-neg-in N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

   5. Applied rewrites 67.4%

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

   if -3.99999999999999997e-299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.99999999999999978e282

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. sub-flip N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 67.4

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

   3. Applied rewrites 67.4%

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

Alternative 3: 87.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := z \cdot \left(t - a\right)\\ t_4 := \frac{x \cdot y + t\_3}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{t\_2}, z, \frac{x}{y} \cdot y\right)\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t\_3\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (fma (- b y) z y))
        (t_3 (* z (- t a)))
        (t_4 (/ (+ (* x y) t_3) (+ y (* z (- b y))))))
   (if (<= t_4 (- INFINITY))
     (fma (/ (- t a) t_2) z (* (/ x y) y))
     (if (<= t_4 -4e-299)
       (/ (fma x y t_3) (fma z (- b y) y))
       (if (<= t_4 0.0)
         t_1
         (if (<= t_4 2e+267) (/ (fma (- t a) z (* y x)) t_2) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = fma((b - y), z, y);
	double t_3 = z * (t - a);
	double t_4 = ((x * y) + t_3) / (y + (z * (b - y)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = fma(((t - a) / t_2), z, ((x / y) * y));
	} else if (t_4 <= -4e-299) {
		tmp = fma(x, y, t_3) / fma(z, (b - y), y);
	} else if (t_4 <= 0.0) {
		tmp = t_1;
	} else if (t_4 <= 2e+267) {
		tmp = fma((t - a), z, (y * x)) / t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(z * Float64(t - a))
	t_4 = Float64(Float64(Float64(x * y) + t_3) / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = fma(Float64(Float64(t - a) / t_2), z, Float64(Float64(x / y) * y));
	elseif (t_4 <= -4e-299)
		tmp = Float64(fma(x, y, t_3) / fma(z, Float64(b - y), y));
	elseif (t_4 <= 0.0)
		tmp = t_1;
	elseif (t_4 <= 2e+267)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / 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[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * y), $MachinePrecision] + t$95$3), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(N[(x / y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -4e-299], N[(N[(x * y + t$95$3), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], t$95$1, If[LessEqual[t$95$4, 2e+267], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

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

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

      17. distribute-rgt-neg-in N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 51.4

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

   6. Applied rewrites 51.4%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -3.99999999999999997e-299

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

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

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

      17. distribute-rgt-neg-in N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

   5. Applied rewrites 67.4%

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

   if -3.99999999999999997e-299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or 1.9999999999999999e267 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e267

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. sub-flip N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 67.4

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

   3. Applied rewrites 67.4%

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

Alternative 4: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -5.7 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-174}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(\frac{z}{x}, t - a, y\right) \cdot \frac{1}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -5.7e+82)
     t_1
     (if (<= z 3.7e-174)
       (* x (* (fma (/ z x) (- t a) y) (/ 1.0 (fma z (- b y) y))))
       (if (<= z 2.8e+75)
         (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b 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 <= -5.7e+82) {
		tmp = t_1;
	} else if (z <= 3.7e-174) {
		tmp = x * (fma((z / x), (t - a), y) * (1.0 / fma(z, (b - y), y)));
	} else if (z <= 2.8e+75) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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 <= -5.7e+82)
		tmp = t_1;
	elseif (z <= 3.7e-174)
		tmp = Float64(x * Float64(fma(Float64(z / x), Float64(t - a), y) * Float64(1.0 / fma(z, Float64(b - y), y))));
	elseif (z <= 2.8e+75)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - 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, -5.7e+82], t$95$1, If[LessEqual[z, 3.7e-174], N[(x * N[(N[(N[(z / x), $MachinePrecision] * N[(t - a), $MachinePrecision] + y), $MachinePrecision] * N[(1.0 / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e+75], 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], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.70000000000000016e82 or 2.80000000000000012e75 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -5.70000000000000016e82 < z < 3.7000000000000001e-174

   1. Initial program 67.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 66.2

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

   4. Applied rewrites 66.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. div-add-rev N/A

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

      8. mult-flip N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 65.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 65.0

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

   6. Applied rewrites 65.0%

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

   if 3.7000000000000001e-174 < z < 2.80000000000000012e75

   1. Initial program 67.4%

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

Alternative 5: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -8 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -8e+58)
     t_1
     (if (<= z 2.8e+75) (/ (fma y x (* (- t a) z)) (+ y (* z (- b 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 <= -8e+58) {
		tmp = t_1;
	} else if (z <= 2.8e+75) {
		tmp = fma(y, x, ((t - a) * z)) / (y + (z * (b - 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 <= -8e+58)
		tmp = t_1;
	elseif (z <= 2.8e+75)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(y + Float64(z * Float64(b - 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, -8e+58], t$95$1, If[LessEqual[z, 2.8e+75], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999955e58 or 2.80000000000000012e75 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -7.99999999999999955e58 < z < 2.80000000000000012e75

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. 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

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

      4. lower-fma.f64 67.4

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.4

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

   3. Applied rewrites 67.4%

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

Alternative 6: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -8 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -8e+58)
     t_1
     (if (<= z 2.8e+75)
       (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b 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 <= -8e+58) {
		tmp = t_1;
	} else if (z <= 2.8e+75) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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 <= (-8d+58)) then
        tmp = t_1
    else if (z <= 2.8d+75) then
        tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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 <= -8e+58) {
		tmp = t_1;
	} else if (z <= 2.8e+75) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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 <= -8e+58:
		tmp = t_1
	elif z <= 2.8e+75:
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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 <= -8e+58)
		tmp = t_1;
	elseif (z <= 2.8e+75)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - 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 <= -8e+58)
		tmp = t_1;
	elseif (z <= 2.8e+75)
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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, -8e+58], t$95$1, If[LessEqual[z, 2.8e+75], 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], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999955e58 or 2.80000000000000012e75 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -7.99999999999999955e58 < z < 2.80000000000000012e75

   1. Initial program 67.4%

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

Alternative 7: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -8 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -8e+58)
     t_1
     (if (<= z 2.8e+75) (/ (fma (- t a) z (* y x)) (fma (- b y) z y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -8e+58) {
		tmp = t_1;
	} else if (z <= 2.8e+75) {
		tmp = fma((t - a), z, (y * x)) / fma((b - y), z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -8e+58)
		tmp = t_1;
	elseif (z <= 2.8e+75)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(b - y), z, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8e+58], t$95$1, If[LessEqual[z, 2.8e+75], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999955e58 or 2.80000000000000012e75 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -7.99999999999999955e58 < z < 2.80000000000000012e75

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. sub-flip N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 67.4

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

   3. Applied rewrites 67.4%

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

Alternative 8: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -8 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -8e+58)
     t_1
     (if (<= z 2.8e+75) (/ (fma x y (* z (- t a))) (fma z (- b 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 <= -8e+58) {
		tmp = t_1;
	} else if (z <= 2.8e+75) {
		tmp = fma(x, y, (z * (t - a))) / fma(z, (b - 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 <= -8e+58)
		tmp = t_1;
	elseif (z <= 2.8e+75)
		tmp = Float64(fma(x, y, Float64(z * Float64(t - a))) / fma(z, Float64(b - 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, -8e+58], t$95$1, If[LessEqual[z, 2.8e+75], N[(N[(x * y + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999955e58 or 2.80000000000000012e75 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -7.99999999999999955e58 < z < 2.80000000000000012e75

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

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

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

      17. distribute-rgt-neg-in N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

   5. Applied rewrites 67.4%

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

Alternative 9: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.1 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.1e+41)
     t_1
     (if (<= z 2.45e-10) (/ (fma y x (* (- t a) z)) (+ y (* z b))) 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 <= -6.1e+41) {
		tmp = t_1;
	} else if (z <= 2.45e-10) {
		tmp = fma(y, x, ((t - a) * z)) / (y + (z * b));
	} 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 <= -6.1e+41)
		tmp = t_1;
	elseif (z <= 2.45e-10)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(y + Float64(z * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.1e+41], t$95$1, If[LessEqual[z, 2.45e-10], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.09999999999999998e41 or 2.4499999999999998e-10 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -6.09999999999999998e41 < z < 2.4499999999999998e-10

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. 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

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

      4. lower-fma.f64 67.4

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.4

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

   3. Applied rewrites 67.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 58.0%

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

Alternative 10: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.1 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.1e+41)
     t_1
     (if (<= z 2.45e-10) (/ (fma x y (* z (- t a))) (fma z b 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 <= -6.1e+41) {
		tmp = t_1;
	} else if (z <= 2.45e-10) {
		tmp = fma(x, y, (z * (t - a))) / fma(z, b, 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 <= -6.1e+41)
		tmp = t_1;
	elseif (z <= 2.45e-10)
		tmp = Float64(fma(x, y, Float64(z * Float64(t - a))) / fma(z, b, 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, -6.1e+41], t$95$1, If[LessEqual[z, 2.45e-10], N[(N[(x * y + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * b + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.09999999999999998e41 or 2.4499999999999998e-10 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -6.09999999999999998e41 < z < 2.4499999999999998e-10

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

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

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

      17. distribute-rgt-neg-in N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.0%

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

Alternative 11: 71.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -7 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+47}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -7e+58)
     t_1
     (if (<= z 1.15e+47) (/ (fma t z (* x y)) (fma z (- b 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 <= -7e+58) {
		tmp = t_1;
	} else if (z <= 1.15e+47) {
		tmp = fma(t, z, (x * y)) / fma(z, (b - 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 <= -7e+58)
		tmp = t_1;
	elseif (z <= 1.15e+47)
		tmp = Float64(fma(t, z, Float64(x * y)) / fma(z, Float64(b - 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, -7e+58], t$95$1, If[LessEqual[z, 1.15e+47], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.9999999999999995e58 or 1.1499999999999999e47 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -6.9999999999999995e58 < z < 1.1499999999999999e47

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

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

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

      17. distribute-rgt-neg-in N/A

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

      18. lift-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 48.5

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

   8. Applied rewrites 48.5%

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

Alternative 12: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -16000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-140}:\\ \;\;\;\;x \cdot \frac{y}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(t - a, z, y \cdot x\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -16000000.0)
     t_1
     (if (<= z -8e-140)
       (* x (/ y (+ y (* z (- b y)))))
       (if (<= z 7.4e-12) (* (fma (- t a) z (* y x)) (/ 1.0 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 <= -16000000.0) {
		tmp = t_1;
	} else if (z <= -8e-140) {
		tmp = x * (y / (y + (z * (b - y))));
	} else if (z <= 7.4e-12) {
		tmp = fma((t - a), z, (y * x)) * (1.0 / 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 <= -16000000.0)
		tmp = t_1;
	elseif (z <= -8e-140)
		tmp = Float64(x * Float64(y / Float64(y + Float64(z * Float64(b - y)))));
	elseif (z <= 7.4e-12)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) * Float64(1.0 / 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, -16000000.0], t$95$1, If[LessEqual[z, -8e-140], N[(x * N[(y / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.4e-12], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6e7 or 7.39999999999999997e-12 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -1.6e7 < z < -7.9999999999999999e-140

   1. Initial program 67.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 36.6

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

   7. Applied rewrites 36.6%

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

   if -7.9999999999999999e-140 < z < 7.39999999999999997e-12

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 67.3

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. add-flip N/A

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

      16. sub-flip N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. remove-double-neg N/A

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

      20. lower-fma.f64 67.3

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

   3. Applied rewrites 67.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 33.2

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

   6. Applied rewrites 33.2%

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

Alternative 13: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -16000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;x \cdot \frac{y}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -16000000.0)
     t_1
     (if (<= z 4.8e-148)
       (* x (/ y (+ y (* z (- b y)))))
       (if (<= z 4.8e+14) (/ (* (- 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 <= -16000000.0) {
		tmp = t_1;
	} else if (z <= 4.8e-148) {
		tmp = x * (y / (y + (z * (b - y))));
	} else if (z <= 4.8e+14) {
		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 <= -16000000.0)
		tmp = t_1;
	elseif (z <= 4.8e-148)
		tmp = Float64(x * Float64(y / Float64(y + Float64(z * Float64(b - y)))));
	elseif (z <= 4.8e+14)
		tmp = Float64(Float64(Float64(t - a) * 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, -16000000.0], t$95$1, If[LessEqual[z, 4.8e-148], N[(x * N[(y / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.8e+14], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6e7 or 4.8e14 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -1.6e7 < z < 4.8000000000000002e-148

   1. Initial program 67.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 36.6

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

   7. Applied rewrites 36.6%

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

   if 4.8000000000000002e-148 < z < 4.8e14

   1. Initial program 67.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 42.5

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

   4. Applied rewrites 42.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.5

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

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + z \cdot \left(b - y\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(+-commutative, \left(z \cdot \left(b - y\right) + y\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(add-flip, \left(z \cdot \left(b - y\right) - \left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(sub-flip, \left(z \cdot \left(b - y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(z \cdot \left(b - y\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{z \cdot \mathsf{Rewrite=>}\left(lift--.f64, \left(b - y\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{z \cdot \mathsf{Rewrite=>}\left(sub-negate-rev, \left(\mathsf{neg}\left(\left(y - b\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(distribute-rgt-neg-out, \left(\mathsf{neg}\left(z \cdot \left(y - b\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(distribute-lft-neg-out, \left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(y - b\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(y - b\right) + \mathsf{Rewrite=>}\left(remove-double-neg, y\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(*-commutative, \left(\left(y - b\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)\right) + y} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(distribute-rgt-neg-in, \left(\mathsf{neg}\left(\left(y - b\right) \cdot z\right)\right)\right) + y} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(distribute-lft-neg-out, \left(\left(\mathsf{neg}\left(\left(y - b\right)\right)\right) \cdot z\right)\right) + y} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(sub-negate-rev, \left(b - y\right)\right) \cdot z + y} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(b - y\right)\right) \cdot z + y} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(b - y, z, y\right)\right)\right)} \]

   6. Applied rewrites 42.5%

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

Alternative 14: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -16000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;x \cdot \frac{y}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{y + z \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -16000000.0)
     t_1
     (if (<= z 4.8e-148)
       (* x (/ y (+ y (* z (- b y)))))
       (if (<= z 2.4e-15) (/ (* z (- t a)) (+ y (* z b))) 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 <= -16000000.0) {
		tmp = t_1;
	} else if (z <= 4.8e-148) {
		tmp = x * (y / (y + (z * (b - y))));
	} else if (z <= 2.4e-15) {
		tmp = (z * (t - a)) / (y + (z * 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 - a) / (b - y)
    if (z <= (-16000000.0d0)) then
        tmp = t_1
    else if (z <= 4.8d-148) then
        tmp = x * (y / (y + (z * (b - y))))
    else if (z <= 2.4d-15) then
        tmp = (z * (t - a)) / (y + (z * 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 - a) / (b - y);
	double tmp;
	if (z <= -16000000.0) {
		tmp = t_1;
	} else if (z <= 4.8e-148) {
		tmp = x * (y / (y + (z * (b - y))));
	} else if (z <= 2.4e-15) {
		tmp = (z * (t - a)) / (y + (z * b));
	} 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 <= -16000000.0:
		tmp = t_1
	elif z <= 4.8e-148:
		tmp = x * (y / (y + (z * (b - y))))
	elif z <= 2.4e-15:
		tmp = (z * (t - a)) / (y + (z * b))
	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 <= -16000000.0)
		tmp = t_1;
	elseif (z <= 4.8e-148)
		tmp = Float64(x * Float64(y / Float64(y + Float64(z * Float64(b - y)))));
	elseif (z <= 2.4e-15)
		tmp = Float64(Float64(z * Float64(t - a)) / Float64(y + Float64(z * b)));
	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 <= -16000000.0)
		tmp = t_1;
	elseif (z <= 4.8e-148)
		tmp = x * (y / (y + (z * (b - y))));
	elseif (z <= 2.4e-15)
		tmp = (z * (t - a)) / (y + (z * b));
	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, -16000000.0], t$95$1, If[LessEqual[z, 4.8e-148], N[(x * N[(y / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.4e-15], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6e7 or 2.39999999999999995e-15 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -1.6e7 < z < 4.8000000000000002e-148

   1. Initial program 67.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 36.6

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

   7. Applied rewrites 36.6%

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

   if 4.8000000000000002e-148 < z < 2.39999999999999995e-15

   1. Initial program 67.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 42.5

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

   4. Applied rewrites 42.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 35.6%

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

Alternative 15: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -16000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -16000000.0)
     t_1
     (if (<= z 6.2e-13) (* x (/ y (+ y (* z (- b 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 <= -16000000.0) {
		tmp = t_1;
	} else if (z <= 6.2e-13) {
		tmp = x * (y / (y + (z * (b - 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 <= (-16000000.0d0)) then
        tmp = t_1
    else if (z <= 6.2d-13) then
        tmp = x * (y / (y + (z * (b - 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 <= -16000000.0) {
		tmp = t_1;
	} else if (z <= 6.2e-13) {
		tmp = x * (y / (y + (z * (b - 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 <= -16000000.0:
		tmp = t_1
	elif z <= 6.2e-13:
		tmp = x * (y / (y + (z * (b - 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 <= -16000000.0)
		tmp = t_1;
	elseif (z <= 6.2e-13)
		tmp = Float64(x * Float64(y / Float64(y + Float64(z * Float64(b - 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 <= -16000000.0)
		tmp = t_1;
	elseif (z <= 6.2e-13)
		tmp = x * (y / (y + (z * (b - 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, -16000000.0], t$95$1, If[LessEqual[z, 6.2e-13], N[(x * N[(y / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6e7 or 6.1999999999999998e-13 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -1.6e7 < z < 6.1999999999999998e-13

   1. Initial program 67.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 36.6

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

   7. Applied rewrites 36.6%

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

Alternative 16: 64.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-73}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1e-55) t_1 (if (<= z 6.5e-73) (* x 1.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1e-55) {
		tmp = t_1;
	} else if (z <= 6.5e-73) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-1d-55)) then
        tmp = t_1
    else if (z <= 6.5d-73) then
        tmp = x * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1e-55) {
		tmp = t_1;
	} else if (z <= 6.5e-73) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -1e-55:
		tmp = t_1
	elif z <= 6.5e-73:
		tmp = x * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1e-55)
		tmp = t_1;
	elseif (z <= 6.5e-73)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -1e-55)
		tmp = t_1;
	elseif (z <= 6.5e-73)
		tmp = x * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e-55], t$95$1, If[LessEqual[z, 6.5e-73], N[(x * 1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999995e-56 or 6.4999999999999999e-73 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -9.99999999999999995e-56 < z < 6.4999999999999999e-73

   1. Initial program 67.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 25.6%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 48.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -650:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;y \leq 54000000000:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -650.0)
   (* x 1.0)
   (if (<= y 54000000000.0) (/ (- t a) b) (* x 1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -650.0) {
		tmp = x * 1.0;
	} else if (y <= 54000000000.0) {
		tmp = (t - a) / b;
	} else {
		tmp = x * 1.0;
	}
	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 <= (-650.0d0)) then
        tmp = x * 1.0d0
    else if (y <= 54000000000.0d0) then
        tmp = (t - a) / b
    else
        tmp = x * 1.0d0
    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 <= -650.0) {
		tmp = x * 1.0;
	} else if (y <= 54000000000.0) {
		tmp = (t - a) / b;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -650.0:
		tmp = x * 1.0
	elif y <= 54000000000.0:
		tmp = (t - a) / b
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -650.0)
		tmp = Float64(x * 1.0);
	elseif (y <= 54000000000.0)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -650.0)
		tmp = x * 1.0;
	elseif (y <= 54000000000.0)
		tmp = (t - a) / b;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -650.0], N[(x * 1.0), $MachinePrecision], If[LessEqual[y, 54000000000.0], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -650 or 5.4e10 < y

   1. Initial program 67.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 25.6%

      \[\leadsto x \cdot 1 \]

   if -650 < y < 5.4e10

   1. Initial program 67.4%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 35.5

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

   4. Applied rewrites 35.5%

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

Alternative 18: 37.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-8}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.2e-22) (/ t b) (if (<= z 4.1e-8) (* x 1.0) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.2e-22) {
		tmp = t / b;
	} else if (z <= 4.1e-8) {
		tmp = x * 1.0;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-4.2d-22)) then
        tmp = t / b
    else if (z <= 4.1d-8) then
        tmp = x * 1.0d0
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.2e-22) {
		tmp = t / b;
	} else if (z <= 4.1e-8) {
		tmp = x * 1.0;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -4.2e-22:
		tmp = t / b
	elif z <= 4.1e-8:
		tmp = x * 1.0
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.2e-22)
		tmp = Float64(t / b);
	elseif (z <= 4.1e-8)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -4.2e-22)
		tmp = t / b;
	elseif (z <= 4.1e-8)
		tmp = x * 1.0;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.2e-22], N[(t / b), $MachinePrecision], If[LessEqual[z, 4.1e-8], N[(x * 1.0), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.20000000000000016e-22 or 4.10000000000000032e-8 < z

   1. Initial program 67.4%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 35.5

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 20.6

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

   7. Applied rewrites 20.6%

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

   if -4.20000000000000016e-22 < z < 4.10000000000000032e-8

   1. Initial program 67.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 25.6%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 25.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

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

C

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

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 * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 67.4%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 66.2

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

4. Applied rewrites 66.2%

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

5. Taylor expanded in z around 0

   \[\leadsto x \cdot 1 \]
6. Step-by-step derivation

7. Applied rewrites 25.6%

   \[\leadsto x \cdot 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64
  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))