Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.5% → 91.0%

Time: 4.4s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% 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: 91.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := x \cdot y + z \cdot \left(t - a\right)\\ t_4 := \frac{t\_3}{y + z \cdot \left(b - y\right)}\\ t_5 := \frac{t - a}{t\_2}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_5, z, \frac{x}{t\_2} \cdot y\right)\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(b, z, y\right) - z \cdot y}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t\_2}, x, t\_5 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (fma (- b y) z y))
        (t_3 (+ (* x y) (* z (- t a))))
        (t_4 (/ t_3 (+ y (* z (- b y)))))
        (t_5 (/ (- t a) t_2)))
   (if (<= t_4 (- INFINITY))
     (fma t_5 z (* (/ x t_2) y))
     (if (<= t_4 -1e-285)
       (/ t_3 (- (fma b z y) (* z y)))
       (if (<= t_4 0.0)
         t_1
         (if (<= t_4 INFINITY) (fma (/ y t_2) x (* t_5 z)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = fma((b - y), z, y);
	double t_3 = (x * y) + (z * (t - a));
	double t_4 = t_3 / (y + (z * (b - y)));
	double t_5 = (t - a) / t_2;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = fma(t_5, z, ((x / t_2) * y));
	} else if (t_4 <= -1e-285) {
		tmp = t_3 / (fma(b, z, y) - (z * y));
	} else if (t_4 <= 0.0) {
		tmp = t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = fma((y / t_2), x, (t_5 * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(x * y) + Float64(z * Float64(t - a)))
	t_4 = Float64(t_3 / Float64(y + Float64(z * Float64(b - y))))
	t_5 = Float64(Float64(t - a) / t_2)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = fma(t_5, z, Float64(Float64(x / t_2) * y));
	elseif (t_4 <= -1e-285)
		tmp = Float64(t_3 / Float64(fma(b, z, y) - Float64(z * y)));
	elseif (t_4 <= 0.0)
		tmp = t_1;
	elseif (t_4 <= Inf)
		tmp = fma(Float64(y / t_2), x, Float64(t_5 * z));
	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[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(t$95$5 * z + N[(N[(x / t$95$2), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-285], N[(t$95$3 / N[(N[(b * z + y), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], t$95$1, If[LessEqual[t$95$4, Infinity], N[(N[(y / t$95$2), $MachinePrecision] * x + N[(t$95$5 * z), $MachinePrecision]), $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 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

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

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

      7. div-add N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. lift-*.f64 N/A

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

      14. remove-double-neg N/A

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

      15. *-rgt-identity N/A

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

      16. associate-*r/ N/A

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

      17. mult-flip N/A

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

   3. Applied rewrites 69.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-285

   1. Initial program 66.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. distribute-lft-in N/A

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

      11. associate--r+ N/A

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

      12. lower--.f64 N/A

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

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

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

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

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

      18. distribute-rgt-neg-out N/A

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

      19. remove-double-neg N/A

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

      20. lower-*.f64 66.4

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

   3. Applied rewrites 66.4%

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

   if -1.00000000000000007e-285 < (/.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 66.5%

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

   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 51.2

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

   4. Applied rewrites 51.2%

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

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

   1. Initial program 66.5%

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

   3. Applied rewrites 75.4%

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

Alternative 2: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := x \cdot y + z \cdot \left(t - a\right)\\ t_3 := \frac{t\_2}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(b - y, z, y\right)\\ t_5 := \mathsf{fma}\left(\frac{y}{t\_4}, x, \frac{t - a}{t\_4} \cdot z\right)\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+304}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(b, z, y\right) - z \cdot y}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (+ (* x y) (* z (- t a))))
        (t_3 (/ t_2 (+ y (* z (- b y)))))
        (t_4 (fma (- b y) z y))
        (t_5 (fma (/ y t_4) x (* (/ (- t a) t_4) z))))
   (if (<= t_3 -1e+304)
     t_5
     (if (<= t_3 -1e-285)
       (/ t_2 (- (fma b z y) (* z y)))
       (if (<= t_3 0.0) t_1 (if (<= t_3 INFINITY) t_5 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = (x * y) + (z * (t - a));
	double t_3 = t_2 / (y + (z * (b - y)));
	double t_4 = fma((b - y), z, y);
	double t_5 = fma((y / t_4), x, (((t - a) / t_4) * z));
	double tmp;
	if (t_3 <= -1e+304) {
		tmp = t_5;
	} else if (t_3 <= -1e-285) {
		tmp = t_2 / (fma(b, z, y) - (z * y));
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(x * y) + Float64(z * Float64(t - a)))
	t_3 = Float64(t_2 / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(Float64(b - y), z, y)
	t_5 = fma(Float64(y / t_4), x, Float64(Float64(Float64(t - a) / t_4) * z))
	tmp = 0.0
	if (t_3 <= -1e+304)
		tmp = t_5;
	elseif (t_3 <= -1e-285)
		tmp = Float64(t_2 / Float64(fma(b, z, y) - Float64(z * y)));
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= Inf)
		tmp = t_5;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$5 = N[(N[(y / t$95$4), $MachinePrecision] * x + N[(N[(N[(t - a), $MachinePrecision] / t$95$4), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+304], t$95$5, If[LessEqual[t$95$3, -1e-285], N[(t$95$2 / N[(N[(b * z + y), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, Infinity], t$95$5, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999994e303 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 66.5%

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

   3. Applied rewrites 75.4%

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

   if -9.9999999999999994e303 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-285

   1. Initial program 66.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. distribute-lft-in N/A

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

      11. associate--r+ N/A

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

      12. lower--.f64 N/A

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

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

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

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

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

      18. distribute-rgt-neg-out N/A

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

      19. remove-double-neg N/A

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

      20. lower-*.f64 66.4

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

   3. Applied rewrites 66.4%

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

   if -1.00000000000000007e-285 < (/.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 66.5%

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

   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 51.2

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

   4. Applied rewrites 51.2%

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

Alternative 3: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := x \cdot y + z \cdot \left(t - a\right)\\ t_3 := \frac{t\_2}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(b - y, z, y\right)\\ t_5 := \mathsf{fma}\left(1, x, \frac{t - a}{t\_4} \cdot z\right)\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+304}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(b, z, y\right) - z \cdot y}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{t\_4}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (+ (* x y) (* z (- t a))))
        (t_3 (/ t_2 (+ y (* z (- b y)))))
        (t_4 (fma (- b y) z y))
        (t_5 (fma 1.0 x (* (/ (- t a) t_4) z))))
   (if (<= t_3 -1e+304)
     t_5
     (if (<= t_3 -1e-285)
       (/ t_2 (- (fma b z y) (* z y)))
       (if (<= t_3 0.0)
         t_1
         (if (<= t_3 1e+287)
           (/ (fma (- t a) z (* y x)) t_4)
           (if (<= t_3 INFINITY) t_5 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = (x * y) + (z * (t - a));
	double t_3 = t_2 / (y + (z * (b - y)));
	double t_4 = fma((b - y), z, y);
	double t_5 = fma(1.0, x, (((t - a) / t_4) * z));
	double tmp;
	if (t_3 <= -1e+304) {
		tmp = t_5;
	} else if (t_3 <= -1e-285) {
		tmp = t_2 / (fma(b, z, y) - (z * y));
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= 1e+287) {
		tmp = fma((t - a), z, (y * x)) / t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(x * y) + Float64(z * Float64(t - a)))
	t_3 = Float64(t_2 / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(Float64(b - y), z, y)
	t_5 = fma(1.0, x, Float64(Float64(Float64(t - a) / t_4) * z))
	tmp = 0.0
	if (t_3 <= -1e+304)
		tmp = t_5;
	elseif (t_3 <= -1e-285)
		tmp = Float64(t_2 / Float64(fma(b, z, y) - Float64(z * y)));
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= 1e+287)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / t_4);
	elseif (t_3 <= Inf)
		tmp = t_5;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 * x + N[(N[(N[(t - a), $MachinePrecision] / t$95$4), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+304], t$95$5, If[LessEqual[t$95$3, -1e-285], N[(t$95$2 / N[(N[(b * z + y), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, 1e+287], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$5, 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)))) < -9.9999999999999994e303 or 1.0000000000000001e287 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 66.5%

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 55.8%

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

   if -9.9999999999999994e303 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-285

   1. Initial program 66.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. distribute-lft-in N/A

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

      11. associate--r+ N/A

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

      12. lower--.f64 N/A

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

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

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

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

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

      18. distribute-rgt-neg-out N/A

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

      19. remove-double-neg N/A

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

      20. lower-*.f64 66.4

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

   3. Applied rewrites 66.4%

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

   if -1.00000000000000007e-285 < (/.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 66.5%

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

   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 51.2

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

   4. Applied rewrites 51.2%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.0000000000000001e287

   1. Initial program 66.5%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 66.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. sub-flip N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. remove-double-neg N/A

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

      22. lower-fma.f64 66.5

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

   3. Applied rewrites 66.5%

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

Alternative 4: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(b - y, z, y\right)\\ t_4 := \mathsf{fma}\left(1, x, \frac{t - a}{t\_3} \cdot z\right)\\ t_5 := \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{t\_3}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+304}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+287}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma (- b y) z y))
        (t_4 (fma 1.0 x (* (/ (- t a) t_3) z)))
        (t_5 (/ (fma (- t a) z (* y x)) t_3)))
   (if (<= t_2 -1e+304)
     t_4
     (if (<= t_2 -1e-285)
       t_5
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 1e+287) t_5 (if (<= t_2 INFINITY) t_4 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma((b - y), z, y);
	double t_4 = fma(1.0, x, (((t - a) / t_3) * z));
	double t_5 = fma((t - a), z, (y * x)) / t_3;
	double tmp;
	if (t_2 <= -1e+304) {
		tmp = t_4;
	} else if (t_2 <= -1e-285) {
		tmp = t_5;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+287) {
		tmp = t_5;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(Float64(b - y), z, y)
	t_4 = fma(1.0, x, Float64(Float64(Float64(t - a) / t_3) * z))
	t_5 = Float64(fma(Float64(t - a), z, Float64(y * x)) / t_3)
	tmp = 0.0
	if (t_2 <= -1e+304)
		tmp = t_4;
	elseif (t_2 <= -1e-285)
		tmp = t_5;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 1e+287)
		tmp = t_5;
	elseif (t_2 <= Inf)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 * x + N[(N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+304], t$95$4, If[LessEqual[t$95$2, -1e-285], t$95$5, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 1e+287], t$95$5, If[LessEqual[t$95$2, Infinity], t$95$4, t$95$1]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999994e303 or 1.0000000000000001e287 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 66.5%

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 55.8%

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

   if -9.9999999999999994e303 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000007e-285 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.0000000000000001e287

   1. Initial program 66.5%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 66.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. sub-flip N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. remove-double-neg N/A

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

      22. lower-fma.f64 66.5

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

   3. Applied rewrites 66.5%

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

   if -1.00000000000000007e-285 < (/.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 66.5%

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

   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 51.2

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

   4. Applied rewrites 51.2%

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

Alternative 5: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4800000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3e+35)
     t_1
     (if (<= z 4800000000000.0)
       (/ (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 <= -3e+35) {
		tmp = t_1;
	} else if (z <= 4800000000000.0) {
		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 <= -3e+35)
		tmp = t_1;
	elseif (z <= 4800000000000.0)
		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, -3e+35], t$95$1, If[LessEqual[z, 4800000000000.0], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.99999999999999991e35 or 4.8e12 < z

   1. Initial program 66.5%

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

   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 51.2

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

   4. Applied rewrites 51.2%

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

   if -2.99999999999999991e35 < z < 4.8e12

   1. Initial program 66.5%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 66.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. sub-flip N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. remove-double-neg N/A

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

      22. lower-fma.f64 66.5

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

   3. Applied rewrites 66.5%

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

Alternative 6: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2050:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2600000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2050.0)
     t_1
     (if (<= z 2600000000000.0) (/ (fma (- t a) z (* y x)) (fma b 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 <= -2050.0) {
		tmp = t_1;
	} else if (z <= 2600000000000.0) {
		tmp = fma((t - a), z, (y * x)) / fma(b, 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 <= -2050.0)
		tmp = t_1;
	elseif (z <= 2600000000000.0)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(b, 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, -2050.0], t$95$1, If[LessEqual[z, 2600000000000.0], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(b * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2050 or 2.6e12 < z

   1. Initial program 66.5%

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

   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 51.2

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

   4. Applied rewrites 51.2%

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

   if -2050 < z < 2.6e12

   1. Initial program 66.5%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 66.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. sub-flip N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. remove-double-neg N/A

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

      22. lower-fma.f64 66.5

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

   3. Applied rewrites 66.5%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 57.1%

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

Alternative 7: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right)\\ \mathbf{elif}\;z \leq 2600000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{x}{b}, \frac{t - a}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.2e-12)
     t_1
     (if (<= z 1.4e-66)
       (fma 1.0 x (/ (* z (- t a)) y))
       (if (<= z 2600000000000.0) (fma (/ y z) (/ x b) (/ (- t a) 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 <= -4.2e-12) {
		tmp = t_1;
	} else if (z <= 1.4e-66) {
		tmp = fma(1.0, x, ((z * (t - a)) / y));
	} else if (z <= 2600000000000.0) {
		tmp = fma((y / z), (x / b), ((t - a) / 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 <= -4.2e-12)
		tmp = t_1;
	elseif (z <= 1.4e-66)
		tmp = fma(1.0, x, Float64(Float64(z * Float64(t - a)) / y));
	elseif (z <= 2600000000000.0)
		tmp = fma(Float64(y / z), Float64(x / b), Float64(Float64(t - a) / 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, -4.2e-12], t$95$1, If[LessEqual[z, 1.4e-66], N[(1.0 * x + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2600000000000.0], N[(N[(y / z), $MachinePrecision] * N[(x / b), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.19999999999999988e-12 or 2.6e12 < z

   1. Initial program 66.5%

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

   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 51.2

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

   4. Applied rewrites 51.2%

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

   if -4.19999999999999988e-12 < z < 1.4e-66

   1. Initial program 66.5%

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 37.8

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

   9. Applied rewrites 37.8%

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

   if 1.4e-66 < z < 2.6e12

   1. Initial program 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

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

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

      7. div-add N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. lift-*.f64 N/A

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

      14. remove-double-neg N/A

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

      15. *-rgt-identity N/A

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

      16. associate-*r/ N/A

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

      17. mult-flip N/A

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

   3. Applied rewrites 69.9%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.5

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

   6. Applied rewrites 38.5%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 39.3

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

   9. Applied rewrites 39.3%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. +-commutative N/A

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

       5. associate--l+ N/A

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

       6. lift--.f64 N/A

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

       7. div-add N/A

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

       8. lift--.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-/l* N/A

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

       12. *-commutative N/A

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

       13. associate-/l* N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lift--.f64 N/A

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

       18. lower-/.f64 39.2

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

   11. Applied rewrites 39.2%

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

Alternative 8: 74.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.2e-12)
     t_1
     (if (<= z 8e-21) (fma 1.0 x (/ (* z (- t a)) 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 <= -4.2e-12) {
		tmp = t_1;
	} else if (z <= 8e-21) {
		tmp = fma(1.0, x, ((z * (t - a)) / 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 <= -4.2e-12)
		tmp = t_1;
	elseif (z <= 8e-21)
		tmp = fma(1.0, x, Float64(Float64(z * Float64(t - a)) / 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, -4.2e-12], t$95$1, If[LessEqual[z, 8e-21], N[(1.0 * x + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999988e-12 or 7.99999999999999926e-21 < z

   1. Initial program 66.5%

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

   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 51.2

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

   4. Applied rewrites 51.2%

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

   if -4.19999999999999988e-12 < z < 7.99999999999999926e-21

   1. Initial program 66.5%

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 37.8

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

   9. Applied rewrites 37.8%

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

Alternative 9: 63.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-66}:\\ \;\;\;\;-1 \cdot \frac{x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.15e-17)
     t_1
     (if (<= z 1.52e-66) (* -1.0 (/ x (- z 1.0))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.15e-17) {
		tmp = t_1;
	} else if (z <= 1.52e-66) {
		tmp = -1.0 * (x / (z - 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-1.15d-17)) then
        tmp = t_1
    else if (z <= 1.52d-66) then
        tmp = (-1.0d0) * (x / (z - 1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.15e-17) {
		tmp = t_1;
	} else if (z <= 1.52e-66) {
		tmp = -1.0 * (x / (z - 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -1.15e-17:
		tmp = t_1
	elif z <= 1.52e-66:
		tmp = -1.0 * (x / (z - 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.15e-17)
		tmp = t_1;
	elseif (z <= 1.52e-66)
		tmp = Float64(-1.0 * Float64(x / Float64(z - 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -1.15e-17)
		tmp = t_1;
	elseif (z <= 1.52e-66)
		tmp = -1.0 * (x / (z - 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e-17], t$95$1, If[LessEqual[z, 1.52e-66], N[(-1.0 * N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15000000000000004e-17 or 1.51999999999999995e-66 < z

   1. Initial program 66.5%

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

   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 51.2

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

   4. Applied rewrites 51.2%

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

   if -1.15000000000000004e-17 < z < 1.51999999999999995e-66

   1. Initial program 66.5%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 33.0

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

   4. Applied rewrites 33.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{z - 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{t - a}{b - y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (t - a) / (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 = (t - a) / (b - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

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 51.2

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

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
5. Add Preprocessing

Alternative 11: 34.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{t - a}{b} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

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 34.2

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

4. Applied rewrites 34.2%

   \[\leadsto \color{blue}{\frac{t - a}{b}} \]
5. Add Preprocessing

Alternative 12: 19.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{t}{b} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(t / b), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

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 34.2

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

4. Applied rewrites 34.2%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 19.4

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

7. Applied rewrites 19.4%

   \[\leadsto \frac{t}{\color{blue}{b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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)))))