Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.4% → 93.9%

Time: 5.5s

Alternatives: 15

Speedup: 1.3×

Specification

Math

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

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: 66.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := x \cdot y + z \cdot \left(t - a\right)\\ t_2 := \frac{t\_1}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(-y, z, y\right)\right)}\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \frac{t\_1}{y + z \cdot \left(b - y\right)}\\ t_5 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_3\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-292}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 10^{+224}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* x y) (* z (- t a))))
        (t_2 (/ t_1 (fma z b (fma (- y) z y))))
        (t_3 (/ (- t a) (- b y)))
        (t_4 (/ t_1 (+ y (* z (- b y)))))
        (t_5 (fma (/ -1.0 (- z 1.0)) x t_3)))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -1e-292)
       t_2
       (if (<= t_4 0.0) t_3 (if (<= t_4 1e+224) t_2 t_5))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * y) + (z * (t - a));
	double t_2 = t_1 / fma(z, b, fma(-y, z, y));
	double t_3 = (t - a) / (b - y);
	double t_4 = t_1 / (y + (z * (b - y)));
	double t_5 = fma((-1.0 / (z - 1.0)), x, t_3);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -1e-292) {
		tmp = t_2;
	} else if (t_4 <= 0.0) {
		tmp = t_3;
	} else if (t_4 <= 1e+224) {
		tmp = t_2;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * y) + Float64(z * Float64(t - a)))
	t_2 = Float64(t_1 / fma(z, b, fma(Float64(-y), z, y)))
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	t_4 = Float64(t_1 / Float64(y + Float64(z * Float64(b - y))))
	t_5 = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -1e-292)
		tmp = t_2;
	elseif (t_4 <= 0.0)
		tmp = t_3;
	elseif (t_4 <= 1e+224)
		tmp = t_2;
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(z * b + N[((-y) * z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -1e-292], t$95$2, If[LessEqual[t$95$4, 0.0], t$95$3, If[LessEqual[t$95$4, 1e+224], t$95$2, t$95$5]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 9.9999999999999997e223 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

   3. Applied rewrites 79.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.5%

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 65.9%

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

   9. Applied rewrites 65.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.0000000000000001e-292 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.9999999999999997e223

   1. Initial program 66.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 66.4%

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

   3. Applied rewrites 66.4%

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

   if -1.0000000000000001e-292 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

Alternative 2: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_4 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_2\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (fma (/ -1.0 (- z 1.0)) x t_2)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-292)
       (/ (fma (- t a) z (* y x)) (fma (- b y) z y))
       (if (<= t_3 0.0)
         t_2
         (if (<= t_3 1e+224) (/ (fma y x (* (- t a) z)) t_1) t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = (t - a) / (b - y);
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = fma((-1.0 / (z - 1.0)), x, t_2);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-292) {
		tmp = fma((t - a), z, (y * x)) / fma((b - y), z, y);
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 1e+224) {
		tmp = fma(y, x, ((t - a) * z)) / t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-292)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(b - y), z, y));
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 1e+224)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-292], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 1e+224], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$4]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 9.9999999999999997e223 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

   3. Applied rewrites 79.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.5%

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 65.9%

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

   9. Applied rewrites 65.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.0000000000000001e-292

   1. Initial program 66.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.4%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.4%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.4%

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

   3. Applied rewrites 66.4%

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

   if -1.0000000000000001e-292 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.9999999999999997e223

   1. Initial program 66.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 66.4%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.4%

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

   3. Applied rewrites 66.4%

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

Alternative 3: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_2\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+224}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (- t a) z (* y x)) (fma (- b y) z y)))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma (/ -1.0 (- z 1.0)) x t_2)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-292)
       t_1
       (if (<= t_3 0.0) t_2 (if (<= t_3 1e+224) t_1 t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t - a), z, (y * x)) / fma((b - y), z, y);
	double t_2 = (t - a) / (b - y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma((-1.0 / (z - 1.0)), x, t_2);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-292) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 1e+224) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(b - y), z, y))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-292)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 1e+224)
		tmp = t_1;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-292], t$95$1, If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 1e+224], t$95$1, t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 9.9999999999999997e223 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

   3. Applied rewrites 79.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.5%

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 65.9%

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

   9. Applied rewrites 65.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.0000000000000001e-292 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.9999999999999997e223

   1. Initial program 66.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.4%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.4%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.4%

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

   3. Applied rewrites 66.4%

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

   if -1.0000000000000001e-292 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

Alternative 4: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{y}{t\_1}\\ t_3 := \mathsf{fma}\left(t\_2, x, \frac{t - a}{b - y}\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+36}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, x, \frac{z}{t\_1} \cdot \left(t - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (/ y t_1))
        (t_3 (fma t_2 x (/ (- t a) (- b y)))))
   (if (<= z -4.2e+36)
     t_3
     (if (<= z 8.5e-11) (fma t_2 x (* (/ z t_1) (- t a))) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = y / t_1;
	double t_3 = fma(t_2, x, ((t - a) / (b - y)));
	double tmp;
	if (z <= -4.2e+36) {
		tmp = t_3;
	} else if (z <= 8.5e-11) {
		tmp = fma(t_2, x, ((z / t_1) * (t - a)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(y / t_1)
	t_3 = fma(t_2, x, Float64(Float64(t - a) / Float64(b - y)))
	tmp = 0.0
	if (z <= -4.2e+36)
		tmp = t_3;
	elseif (z <= 8.5e-11)
		tmp = fma(t_2, x, Float64(Float64(z / t_1) * Float64(t - a)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(y / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * x + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e+36], t$95$3, If[LessEqual[z, 8.5e-11], N[(t$95$2 * x + N[(N[(z / t$95$1), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2000000000000001e36 or 8.5000000000000004e-11 < z

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

   3. Applied rewrites 79.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.5%

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

   6. Applied rewrites 76.5%

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

   if -4.2000000000000001e36 < z < 8.5000000000000004e-11

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

   3. Applied rewrites 79.3%

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

Alternative 5: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_1\right)\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.32e-8)
     (fma (/ -1.0 (- z 1.0)) x t_1)
     (if (<= z 4.8e-7) (/ (+ (* x y) (* z (- t a))) (+ y (* z b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.32e-8) {
		tmp = fma((-1.0 / (z - 1.0)), x, t_1);
	} else if (z <= 4.8e-7) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.32e-8)
		tmp = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_1);
	elseif (z <= 4.8e-7)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.32e-8], N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision], If[LessEqual[z, 4.8e-7], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3200000000000001e-8

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

   3. Applied rewrites 79.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.5%

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 65.9%

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

   9. Applied rewrites 65.9%

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

   if -1.3200000000000001e-8 < z < 4.7999999999999996e-7

   1. Initial program 66.4%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 57.2%

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

   if 4.7999999999999996e-7 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

Alternative 6: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_1\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.5e-42)
     (fma (/ -1.0 (- z 1.0)) x t_1)
     (if (<= z 1.15e-8) (/ (fma t z (* x y)) (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 <= -6.5e-42) {
		tmp = fma((-1.0 / (z - 1.0)), x, t_1);
	} else if (z <= 1.15e-8) {
		tmp = fma(t, z, (x * y)) / 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 <= -6.5e-42)
		tmp = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_1);
	elseif (z <= 1.15e-8)
		tmp = Float64(fma(t, z, Float64(x * y)) / 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, -6.5e-42], N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision], If[LessEqual[z, 1.15e-8], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.4999999999999998e-42

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

   3. Applied rewrites 79.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.5%

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 65.9%

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

   9. Applied rewrites 65.9%

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

   if -6.4999999999999998e-42 < z < 1.15e-8

   1. Initial program 66.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.4%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.4%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.4%

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

   3. Applied rewrites 66.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 47.7%

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

   6. Applied rewrites 47.7%

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

   if 1.15e-8 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

Alternative 7: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.5e-42)
     t_1
     (if (<= z 1.15e-8) (/ (fma t z (* x y)) (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 <= -6.5e-42) {
		tmp = t_1;
	} else if (z <= 1.15e-8) {
		tmp = fma(t, z, (x * y)) / 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 <= -6.5e-42)
		tmp = t_1;
	elseif (z <= 1.15e-8)
		tmp = Float64(fma(t, z, Float64(x * y)) / 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, -6.5e-42], t$95$1, If[LessEqual[z, 1.15e-8], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4999999999999998e-42 or 1.15e-8 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

   if -6.4999999999999998e-42 < z < 1.15e-8

   1. Initial program 66.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.4%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.4%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 66.4%

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

   3. Applied rewrites 66.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 47.7%

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

   6. Applied rewrites 47.7%

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

Alternative 8: 68.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, b - y, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.5e-42)
     t_1
     (if (<= z 1.15e-8) (* (/ y (fma z (- b y) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -6.5e-42) {
		tmp = t_1;
	} else if (z <= 1.15e-8) {
		tmp = (y / fma(z, (b - y), y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -6.5e-42)
		tmp = t_1;
	elseif (z <= 1.15e-8)
		tmp = Float64(Float64(y / fma(z, Float64(b - y), y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e-42], t$95$1, If[LessEqual[z, 1.15e-8], N[(N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4999999999999998e-42 or 1.15e-8 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

   if -6.4999999999999998e-42 < z < 1.15e-8

   1. Initial program 66.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 28.6%

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

   4. Applied rewrites 28.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 36.4%

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 36.4%

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

   6. Applied rewrites 36.4%

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

Alternative 9: 68.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -8.2e-45) t_1 (if (<= z 1.05e-30) (fma (/ (- a) y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -8.2e-45) {
		tmp = t_1;
	} else if (z <= 1.05e-30) {
		tmp = fma((-a / y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -8.2e-45)
		tmp = t_1;
	elseif (z <= 1.05e-30)
		tmp = fma(Float64(Float64(-a) / y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.2e-45], t$95$1, If[LessEqual[z, 1.05e-30], N[(N[((-a) / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1999999999999998e-45 or 1.0500000000000001e-30 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

   if -8.1999999999999998e-45 < z < 1.0500000000000001e-30

   1. Initial program 66.4%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 27.0%

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 30.3%

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

   7. Applied rewrites 30.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 30.3%

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 30.3%

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

   9. Applied rewrites 30.3%

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

Alternative 10: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.8e-63) t_1 (if (<= z 1.05e-30) (+ x (* z (/ t 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 <= -3.8e-63) {
		tmp = t_1;
	} else if (z <= 1.05e-30) {
		tmp = x + (z * (t / y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-3.8d-63)) then
        tmp = t_1
    else if (z <= 1.05d-30) then
        tmp = x + (z * (t / y))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.8e-63) {
		tmp = t_1;
	} else if (z <= 1.05e-30) {
		tmp = x + (z * (t / y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -3.8e-63:
		tmp = t_1
	elif z <= 1.05e-30:
		tmp = x + (z * (t / 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 <= -3.8e-63)
		tmp = t_1;
	elseif (z <= 1.05e-30)
		tmp = Float64(x + Float64(z * Float64(t / y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -3.8e-63)
		tmp = t_1;
	elseif (z <= 1.05e-30)
		tmp = x + (z * (t / y));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.8e-63], t$95$1, If[LessEqual[z, 1.05e-30], N[(x + N[(z * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8000000000000002e-63 or 1.0500000000000001e-30 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

   if -3.8000000000000002e-63 < z < 1.0500000000000001e-30

   1. Initial program 66.4%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 27.0%

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 30.3%

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

   7. Applied rewrites 30.3%

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

   8. Taylor expanded in t around inf

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

      1. lower-/.f64 30.5%

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

   10. Applied rewrites 30.5%

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

Alternative 11: 64.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;x + x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -5.4e-106) t_1 (if (<= z 1.05e-30) (+ x (* x 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 tmp;
	if (z <= -5.4e-106) {
		tmp = t_1;
	} else if (z <= 1.05e-30) {
		tmp = x + (x * z);
	} 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 <= (-5.4d-106)) then
        tmp = t_1
    else if (z <= 1.05d-30) then
        tmp = x + (x * z)
    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 <= -5.4e-106) {
		tmp = t_1;
	} else if (z <= 1.05e-30) {
		tmp = x + (x * z);
	} 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 <= -5.4e-106:
		tmp = t_1
	elif z <= 1.05e-30:
		tmp = x + (x * 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))
	tmp = 0.0
	if (z <= -5.4e-106)
		tmp = t_1;
	elseif (z <= 1.05e-30)
		tmp = Float64(x + Float64(x * z));
	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 <= -5.4e-106)
		tmp = t_1;
	elseif (z <= 1.05e-30)
		tmp = x + (x * z);
	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, -5.4e-106], t$95$1, If[LessEqual[z, 1.05e-30], N[(x + N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004e-106 or 1.0500000000000001e-30 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

   if -5.4000000000000004e-106 < z < 1.0500000000000001e-30

   1. Initial program 66.4%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 27.0%

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 25.4%

         \[\leadsto x + x \cdot z \]

   7. Applied rewrites 25.4%

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

Alternative 12: 64.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{1 + -1 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -5.4e-106)
     t_1
     (if (<= z 1.05e-30) (/ x (+ 1.0 (* -1.0 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 tmp;
	if (z <= -5.4e-106) {
		tmp = t_1;
	} else if (z <= 1.05e-30) {
		tmp = x / (1.0 + (-1.0 * z));
	} 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 <= (-5.4d-106)) then
        tmp = t_1
    else if (z <= 1.05d-30) then
        tmp = x / (1.0d0 + ((-1.0d0) * z))
    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 <= -5.4e-106) {
		tmp = t_1;
	} else if (z <= 1.05e-30) {
		tmp = x / (1.0 + (-1.0 * z));
	} 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 <= -5.4e-106:
		tmp = t_1
	elif z <= 1.05e-30:
		tmp = x / (1.0 + (-1.0 * 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))
	tmp = 0.0
	if (z <= -5.4e-106)
		tmp = t_1;
	elseif (z <= 1.05e-30)
		tmp = Float64(x / Float64(1.0 + Float64(-1.0 * z)));
	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 <= -5.4e-106)
		tmp = t_1;
	elseif (z <= 1.05e-30)
		tmp = x / (1.0 + (-1.0 * z));
	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, -5.4e-106], t$95$1, If[LessEqual[z, 1.05e-30], N[(x / N[(1.0 + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004e-106 or 1.0500000000000001e-30 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 51.5%

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

   4. Applied rewrites 51.5%

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

   if -5.4000000000000004e-106 < z < 1.0500000000000001e-30

   1. Initial program 66.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 33.0%

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

   4. Applied rewrites 33.0%

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

Alternative 13: 49.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;x + x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) b)))
   (if (<= z -2.9e-53) t_1 (if (<= z 1.05e-30) (+ x (* x z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / b;
	double tmp;
	if (z <= -2.9e-53) {
		tmp = t_1;
	} else if (z <= 1.05e-30) {
		tmp = x + (x * z);
	} 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
    if (z <= (-2.9d-53)) then
        tmp = t_1
    else if (z <= 1.05d-30) then
        tmp = x + (x * z)
    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;
	double tmp;
	if (z <= -2.9e-53) {
		tmp = t_1;
	} else if (z <= 1.05e-30) {
		tmp = x + (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / b
	tmp = 0
	if z <= -2.9e-53:
		tmp = t_1
	elif z <= 1.05e-30:
		tmp = x + (x * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / b)
	tmp = 0.0
	if (z <= -2.9e-53)
		tmp = t_1;
	elseif (z <= 1.05e-30)
		tmp = Float64(x + Float64(x * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / b;
	tmp = 0.0;
	if (z <= -2.9e-53)
		tmp = t_1;
	elseif (z <= 1.05e-30)
		tmp = x + (x * z);
	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] / b), $MachinePrecision]}, If[LessEqual[z, -2.9e-53], t$95$1, If[LessEqual[z, 1.05e-30], N[(x + N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8999999999999998e-53 or 1.0500000000000001e-30 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 36.1%

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

   4. Applied rewrites 36.1%

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

   if -2.8999999999999998e-53 < z < 1.0500000000000001e-30

   1. Initial program 66.4%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 27.0%

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 25.4%

         \[\leadsto x + x \cdot z \]

   7. Applied rewrites 25.4%

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

Alternative 14: 37.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -220:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;x + x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -220.0) (/ t b) (if (<= z 1.85e-28) (+ x (* x z)) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -220.0) {
		tmp = t / b;
	} else if (z <= 1.85e-28) {
		tmp = x + (x * z);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-220.0d0)) then
        tmp = t / b
    else if (z <= 1.85d-28) then
        tmp = x + (x * z)
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -220.0) {
		tmp = t / b;
	} else if (z <= 1.85e-28) {
		tmp = x + (x * z);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -220.0:
		tmp = t / b
	elif z <= 1.85e-28:
		tmp = x + (x * z)
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -220.0)
		tmp = Float64(t / b);
	elseif (z <= 1.85e-28)
		tmp = Float64(x + Float64(x * z));
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -220.0)
		tmp = t / b;
	elseif (z <= 1.85e-28)
		tmp = x + (x * z);
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -220.0], N[(t / b), $MachinePrecision], If[LessEqual[z, 1.85e-28], N[(x + N[(x * z), $MachinePrecision]), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -220 or 1.8500000000000001e-28 < z

   1. Initial program 66.4%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 36.1%

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

   4. Applied rewrites 36.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 20.7%

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

   7. Applied rewrites 20.7%

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

   if -220 < z < 1.8500000000000001e-28

   1. Initial program 66.4%

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

   2. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 27.0%

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 25.4%

         \[\leadsto x + x \cdot z \]

   7. Applied rewrites 25.4%

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

Alternative 15: 20.7% accurate, 5.0× speedup

Math

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

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 66.4%

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

2. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 36.1%

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

4. Applied rewrites 36.1%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 20.7%

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

7. Applied rewrites 20.7%

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

Reproduce

herbie shell --seed 2025202 
(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)))))