Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B

Percentage Accurate: 75.5% → 92.3%

Time: 5.8s

Alternatives: 16

Speedup: 0.9×

Specification

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

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 + 1.0d0) + ((y * b) / t))
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 + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 75.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

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 + 1.0d0) + ((y * b) / t))
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 + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t\_2}, \frac{x}{a}\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-315}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{t\_2}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+222}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{1}{b}, \frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, a - -1\right)}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (fma a t (fma b y t))))
   (if (<= t_1 (- INFINITY))
     (fma z (/ y t_2) (/ x a))
     (if (<= t_1 -1e-315)
       t_1
       (if (<= t_1 0.0)
         (/ (fma t x (* z y)) t_2)
         (if (<= t_1 2e+222)
           t_1
           (fma z (/ 1.0 b) (/ x (fma (/ y t) b (- a -1.0))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(a, t, fma(b, y, t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(z, (y / t_2), (x / a));
	} else if (t_1 <= -1e-315) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, x, (z * y)) / t_2;
	} else if (t_1 <= 2e+222) {
		tmp = t_1;
	} else {
		tmp = fma(z, (1.0 / b), (x / fma((y / t), b, (a - -1.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = fma(a, t, fma(b, y, t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(z, Float64(y / t_2), Float64(x / a));
	elseif (t_1 <= -1e-315)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(t, x, Float64(z * y)) / t_2);
	elseif (t_1 <= 2e+222)
		tmp = t_1;
	else
		tmp = fma(z, Float64(1.0 / b), Float64(x / fma(Float64(y / t), b, Float64(a - -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(y / t$95$2), $MachinePrecision] + N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-315], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 2e+222], t$95$1, N[(z * N[(1.0 / b), $MachinePrecision] + N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 75.5%

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

   3. Applied rewrites 88.0%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.9999999848168381e-316 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e222

   1. Initial program 75.5%

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

   if -9.9999999848168381e-316 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.6%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 70.3%

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

   if 2.0000000000000001e222 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.5%

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

   3. Applied rewrites 88.0%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 59.1%

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

   6. Applied rewrites 59.1%

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

Alternative 2: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)}, \frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, a - -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) INFINITY)
   (fma z (/ y (fma a t (fma b y t))) (/ x (fma (/ y t) b (- a -1.0))))
   (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= ((double) INFINITY)) {
		tmp = fma(z, (y / fma(a, t, fma(b, y, t))), (x / fma((y / t), b, (a - -1.0))));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t))) <= Inf)
		tmp = fma(z, Float64(y / fma(a, t, fma(b, y, t))), Float64(x / fma(Float64(y / t), b, Float64(a - -1.0))));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(z * N[(y / N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 75.5%

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

   3. Applied rewrites 88.0%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.3%

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

   4. Applied rewrites 34.3%

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

Alternative 3: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a - -1\right)}\\ \mathbf{if}\;t \leq -7 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (/ y t) z x) (fma (/ y t) b (- a -1.0)))))
   (if (<= t -7e-16)
     t_1
     (if (<= t 7.5e-58) (/ (fma t x (* z y)) (fma a t (fma b y t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / t), z, x) / fma((y / t), b, (a - -1.0));
	double tmp;
	if (t <= -7e-16) {
		tmp = t_1;
	} else if (t <= 7.5e-58) {
		tmp = fma(t, x, (z * y)) / fma(a, t, fma(b, y, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / t), z, x) / fma(Float64(y / t), b, Float64(a - -1.0)))
	tmp = 0.0
	if (t <= -7e-16)
		tmp = t_1;
	elseif (t <= 7.5e-58)
		tmp = Float64(fma(t, x, Float64(z * y)) / fma(a, t, fma(b, y, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(N[(y / t), $MachinePrecision] * b + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7e-16], t$95$1, If[LessEqual[t, 7.5e-58], N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.0000000000000003e-16 or 7.5e-58 < t

   1. Initial program 75.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 75.7%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 77.5%

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

      19. lift-+.f64 N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 77.5%

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

   3. Applied rewrites 77.5%

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

   if -7.0000000000000003e-16 < t < 7.5e-58

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.6%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 70.3%

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

Alternative 4: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{y}{a - -1}, \frac{x}{a - -1}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.4e+45)
   (/ (fma (/ z t) y x) (+ 1.0 a))
   (if (<= t 1.45e+27)
     (/ (fma t x (* z y)) (fma a t (fma b y t)))
     (fma (/ z t) (/ y (- a -1.0)) (/ x (- a -1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.4e+45) {
		tmp = fma((z / t), y, x) / (1.0 + a);
	} else if (t <= 1.45e+27) {
		tmp = fma(t, x, (z * y)) / fma(a, t, fma(b, y, t));
	} else {
		tmp = fma((z / t), (y / (a - -1.0)), (x / (a - -1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.4e+45)
		tmp = Float64(fma(Float64(z / t), y, x) / Float64(1.0 + a));
	elseif (t <= 1.45e+27)
		tmp = Float64(fma(t, x, Float64(z * y)) / fma(a, t, fma(b, y, t)));
	else
		tmp = fma(Float64(z / t), Float64(y / Float64(a - -1.0)), Float64(x / Float64(a - -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.4e+45], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e+27], N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(y / N[(a - -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.3999999999999999e45

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 56.5%

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 56.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{1 + a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{1 + a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{1 + a} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{1 + a} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{z}{t} \cdot y} + x}{1 + a} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 56.6%

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

   6. Applied rewrites 56.6%

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

   if -2.3999999999999999e45 < t < 1.4500000000000001e27

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.6%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 70.3%

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

   if 1.4500000000000001e27 < t

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 56.5%

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 56.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{1 + a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. lower-+.f64 56.5%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{1 + a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{1 + a} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t}} \cdot z + x}{1 + a} \]

      8. lower-*.f64 58.5%

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

   6. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{t} \cdot z + x}{1 + a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z + x}}{1 + a} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{t} \cdot z}{1 + a} + \frac{x}{1 + a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z}}{1 + a} + \frac{x}{1 + a} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + a}} + \frac{x}{1 + a} \]

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{a - \color{blue}{-1}}, \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{x}{1 + a}\right)\right)\right) \]

   8. Applied rewrites 59.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 56.1%

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

   10. Applied rewrites 56.1%

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

Alternative 5: 74.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{elif}\;t \leq 39:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{\mathsf{fma}\left(a, t, b \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{y}{a - -1}, \frac{x}{a - -1}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -9e-42)
   (/ (fma (/ z t) y x) (+ 1.0 a))
   (if (<= t 39.0)
     (/ (fma t x (* z y)) (fma a t (* b y)))
     (fma (/ z t) (/ y (- a -1.0)) (/ x (- a -1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -9e-42) {
		tmp = fma((z / t), y, x) / (1.0 + a);
	} else if (t <= 39.0) {
		tmp = fma(t, x, (z * y)) / fma(a, t, (b * y));
	} else {
		tmp = fma((z / t), (y / (a - -1.0)), (x / (a - -1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -9e-42)
		tmp = Float64(fma(Float64(z / t), y, x) / Float64(1.0 + a));
	elseif (t <= 39.0)
		tmp = Float64(fma(t, x, Float64(z * y)) / fma(a, t, Float64(b * y)));
	else
		tmp = fma(Float64(z / t), Float64(y / Float64(a - -1.0)), Float64(x / Float64(a - -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -9e-42], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 39.0], N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(a * t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(y / N[(a - -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.0000000000000002e-42

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 56.5%

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 56.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{1 + a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{1 + a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{1 + a} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{1 + a} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{z}{t} \cdot y} + x}{1 + a} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 56.6%

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

   6. Applied rewrites 56.6%

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

   if -9.0000000000000002e-42 < t < 39

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.6%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 70.3%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 51.7%

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

   6. Applied rewrites 51.7%

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

   if 39 < t

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 56.5%

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 56.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{1 + a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. lower-+.f64 56.5%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{1 + a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{1 + a} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t}} \cdot z + x}{1 + a} \]

      8. lower-*.f64 58.5%

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

   6. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{t} \cdot z + x}{1 + a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z + x}}{1 + a} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{t} \cdot z}{1 + a} + \frac{x}{1 + a}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z}}{1 + a} + \frac{x}{1 + a} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + a}} + \frac{x}{1 + a} \]

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{a - \color{blue}{-1}}, \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{x}{1 + a}\right)\right)\right) \]

   8. Applied rewrites 59.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 56.1%

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

   10. Applied rewrites 56.1%

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

Alternative 6: 74.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{if}\;t \leq -9 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 39:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{\mathsf{fma}\left(a, t, b \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (/ z t) y x) (+ 1.0 a))))
   (if (<= t -9e-42)
     t_1
     (if (<= t 39.0) (/ (fma t x (* z y)) (fma a t (* b y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((z / t), y, x) / (1.0 + a);
	double tmp;
	if (t <= -9e-42) {
		tmp = t_1;
	} else if (t <= 39.0) {
		tmp = fma(t, x, (z * y)) / fma(a, t, (b * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(z / t), y, x) / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -9e-42)
		tmp = t_1;
	elseif (t <= 39.0)
		tmp = Float64(fma(t, x, Float64(z * y)) / fma(a, t, Float64(b * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9e-42], t$95$1, If[LessEqual[t, 39.0], N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(a * t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.0000000000000002e-42 or 39 < t

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 56.5%

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 56.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{1 + a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{1 + a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{1 + a} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{1 + a} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{z}{t} \cdot y} + x}{1 + a} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 56.6%

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

   6. Applied rewrites 56.6%

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

   if -9.0000000000000002e-42 < t < 39

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.6%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 70.3%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 51.7%

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

   6. Applied rewrites 51.7%

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

Alternative 7: 68.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{t}{y}, z\right)}{b}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{b \cdot y}, \frac{z}{b}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.8e+85)
   (/ (fma x (/ t y) z) b)
   (if (<= b 1.9e+34)
     (/ (fma (/ y t) z x) (- a -1.0))
     (fma x (/ t (* b y)) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.8e+85) {
		tmp = fma(x, (t / y), z) / b;
	} else if (b <= 1.9e+34) {
		tmp = fma((y / t), z, x) / (a - -1.0);
	} else {
		tmp = fma(x, (t / (b * y)), (z / b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.8e+85)
		tmp = Float64(fma(x, Float64(t / y), z) / b);
	elseif (b <= 1.9e+34)
		tmp = Float64(fma(Float64(y / t), z, x) / Float64(a - -1.0));
	else
		tmp = fma(x, Float64(t / Float64(b * y)), Float64(z / b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.8e+85], N[(N[(x * N[(t / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[b, 1.9e+34], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(t / N[(b * y), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7999999999999999e85

   1. Initial program 75.5%

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

   3. Applied rewrites 88.0%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.4%

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

   6. Applied rewrites 40.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 41.5%

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

   8. Applied rewrites 41.5%

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

   if -3.7999999999999999e85 < b < 1.9000000000000001e34

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 56.5%

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 56.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{1 + a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{1 + a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{1 + a} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t}} \cdot z + x}{1 + a} \]

      7. lift-fma.f64 58.5%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 58.5%

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

   6. Applied rewrites 58.5%

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

   if 1.9000000000000001e34 < b

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.6%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 70.3%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 30.8%

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

   6. Applied rewrites 30.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. sub-to-fraction N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. sub-flip N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l/ N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

   8. Applied rewrites 39.7%

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

Alternative 8: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(x, \frac{t}{y}, z\right)}{b}\\ \mathbf{if}\;b \leq -3.8 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma x (/ t y) z) b)))
   (if (<= b -3.8e+85)
     t_1
     (if (<= b 1.9e+34) (/ (fma (/ y t) z x) (- a -1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(x, (t / y), z) / b;
	double tmp;
	if (b <= -3.8e+85) {
		tmp = t_1;
	} else if (b <= 1.9e+34) {
		tmp = fma((y / t), z, x) / (a - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(x, Float64(t / y), z) / b)
	tmp = 0.0
	if (b <= -3.8e+85)
		tmp = t_1;
	elseif (b <= 1.9e+34)
		tmp = Float64(fma(Float64(y / t), z, x) / Float64(a - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[(t / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, -3.8e+85], t$95$1, If[LessEqual[b, 1.9e+34], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.7999999999999999e85 or 1.9000000000000001e34 < b

   1. Initial program 75.5%

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

   3. Applied rewrites 88.0%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.4%

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

   6. Applied rewrites 40.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 41.5%

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

   8. Applied rewrites 41.5%

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

   if -3.7999999999999999e85 < b < 1.9000000000000001e34

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 56.5%

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 56.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{1 + a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{1 + a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{1 + a} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t}} \cdot z + x}{1 + a} \]

      7. lift-fma.f64 58.5%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 58.5%

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

   6. Applied rewrites 58.5%

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

Alternative 9: 60.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{a}\\ \mathbf{if}\;a \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-192}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{y}{t} \cdot z + x}{1}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{t}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) a)))
   (if (<= a -5e+27)
     t_1
     (if (<= a -7e-192)
       (/ (+ z (/ (* t x) y)) b)
       (if (<= a 2.4e-33)
         (/ (+ (* (/ y t) z) x) 1.0)
         (if (<= a 1.25e+56) (/ (fma x (/ t y) z) b) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / a;
	double tmp;
	if (a <= -5e+27) {
		tmp = t_1;
	} else if (a <= -7e-192) {
		tmp = (z + ((t * x) / y)) / b;
	} else if (a <= 2.4e-33) {
		tmp = (((y / t) * z) + x) / 1.0;
	} else if (a <= 1.25e+56) {
		tmp = fma(x, (t / y), z) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / a)
	tmp = 0.0
	if (a <= -5e+27)
		tmp = t_1;
	elseif (a <= -7e-192)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	elseif (a <= 2.4e-33)
		tmp = Float64(Float64(Float64(Float64(y / t) * z) + x) / 1.0);
	elseif (a <= 1.25e+56)
		tmp = Float64(fma(x, Float64(t / y), z) / b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[a, -5e+27], t$95$1, If[LessEqual[a, -7e-192], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[a, 2.4e-33], N[(N[(N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[a, 1.25e+56], N[(N[(x * N[(t / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -4.9999999999999998e27 or 1.2500000000000001e56 < a

   1. Initial program 75.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 33.8%

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

   4. Applied rewrites 33.8%

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

   if -4.9999999999999998e27 < a < -7.0000000000000003e-192

   1. Initial program 75.5%

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

   3. Applied rewrites 88.0%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.4%

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

   6. Applied rewrites 40.4%

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

   if -7.0000000000000003e-192 < a < 2.4e-33

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 56.5%

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 56.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z}{t}}}{1 + a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      3. lower-+.f64 56.5%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{1 + a} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t}} + x}{1 + a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot z}}{t} + x}{1 + a} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{t}} \cdot z + x}{1 + a} \]

      8. lower-*.f64 58.5%

         \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z} + x}{1 + a} \]

   6. Applied rewrites 58.5%

      \[\leadsto \frac{\color{blue}{\frac{y}{t} \cdot z + x}}{1 + a} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{y}{t} \cdot z + x}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 27.4%

      \[\leadsto \frac{\frac{y}{t} \cdot z + x}{1} \]

   if 2.4e-33 < a < 1.2500000000000001e56

   1. Initial program 75.5%

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

   3. Applied rewrites 88.0%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.4%

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

   6. Applied rewrites 40.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 41.5%

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

   8. Applied rewrites 41.5%

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

Alternative 10: 60.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x}{1 + a}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.46:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 a))))
   (if (<= t -8.5e+38) t_1 (if (<= t 0.46) (/ (+ z (/ (* t x) y)) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -8.5e+38) {
		tmp = t_1;
	} else if (t <= 0.46) {
		tmp = (z + ((t * x) / y)) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 + a)
    if (t <= (-8.5d+38)) then
        tmp = t_1
    else if (t <= 0.46d0) then
        tmp = (z + ((t * x) / y)) / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -8.5e+38) {
		tmp = t_1;
	} else if (t <= 0.46) {
		tmp = (z + ((t * x) / y)) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 + a)
	tmp = 0
	if t <= -8.5e+38:
		tmp = t_1
	elif t <= 0.46:
		tmp = (z + ((t * x) / y)) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -8.5e+38)
		tmp = t_1;
	elseif (t <= 0.46)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 + a);
	tmp = 0.0;
	if (t <= -8.5e+38)
		tmp = t_1;
	elseif (t <= 0.46)
		tmp = (z + ((t * x) / y)) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e+38], t$95$1, If[LessEqual[t, 0.46], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.4999999999999997e38 or 0.46000000000000002 < t

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 41.8%

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -8.4999999999999997e38 < t < 0.46000000000000002

   1. Initial program 75.5%

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

   3. Applied rewrites 88.0%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.4%

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

   6. Applied rewrites 40.4%

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

Alternative 11: 59.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x}{1 + a}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.46:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{t}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 a))))
   (if (<= t -8.5e+38) t_1 (if (<= t 0.46) (/ (fma x (/ t y) z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -8.5e+38) {
		tmp = t_1;
	} else if (t <= 0.46) {
		tmp = fma(x, (t / y), z) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -8.5e+38)
		tmp = t_1;
	elseif (t <= 0.46)
		tmp = Float64(fma(x, Float64(t / y), z) / b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e+38], t$95$1, If[LessEqual[t, 0.46], N[(N[(x * N[(t / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.4999999999999997e38 or 0.46000000000000002 < t

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 41.8%

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -8.4999999999999997e38 < t < 0.46000000000000002

   1. Initial program 75.5%

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

   3. Applied rewrites 88.0%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.4%

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

   6. Applied rewrites 40.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 41.5%

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

   8. Applied rewrites 41.5%

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

Alternative 12: 57.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{x}{1 + a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-315}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\frac{z}{b \cdot t} \cdot t\\ \mathbf{elif}\;t\_1 \leq 10^{+291}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (/ x (+ 1.0 a))))
   (if (<= t_1 (- INFINITY))
     (/ z b)
     (if (<= t_1 -1e-315)
       t_2
       (if (<= t_1 2e-310)
         (* (/ z (* b t)) t)
         (if (<= t_1 1e+291) t_2 (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = x / (1.0 + a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_1 <= -1e-315) {
		tmp = t_2;
	} else if (t_1 <= 2e-310) {
		tmp = (z / (b * t)) * t;
	} else if (t_1 <= 1e+291) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = x / (1.0 + a);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = z / b;
	} else if (t_1 <= -1e-315) {
		tmp = t_2;
	} else if (t_1 <= 2e-310) {
		tmp = (z / (b * t)) * t;
	} else if (t_1 <= 1e+291) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	t_2 = x / (1.0 + a)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = z / b
	elif t_1 <= -1e-315:
		tmp = t_2
	elif t_1 <= 2e-310:
		tmp = (z / (b * t)) * t
	elif t_1 <= 1e+291:
		tmp = t_2
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_1 <= -1e-315)
		tmp = t_2;
	elseif (t_1 <= 2e-310)
		tmp = Float64(Float64(z / Float64(b * t)) * t);
	elseif (t_1 <= 1e+291)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	t_2 = x / (1.0 + a);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = z / b;
	elseif (t_1 <= -1e-315)
		tmp = t_2;
	elseif (t_1 <= 2e-310)
		tmp = (z / (b * t)) * t;
	elseif (t_1 <= 1e+291)
		tmp = t_2;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -1e-315], t$95$2, If[LessEqual[t$95$1, 2e-310], N[(N[(z / N[(b * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 1e+291], t$95$2, N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.9999999999999996e290 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -9.9999999848168381e-316 or 1.9999999999999939e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999996e290

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 41.8%

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -9.9999999848168381e-316 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999939e-310

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 62.2%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 28.3%

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

   6. Applied rewrites 28.3%

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

Alternative 13: 55.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+15}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.65e+15) (/ z b) (if (<= y 2.4e-48) (/ x (+ 1.0 a)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.65e+15) {
		tmp = z / b;
	} else if (y <= 2.4e-48) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.65d+15)) then
        tmp = z / b
    else if (y <= 2.4d-48) then
        tmp = x / (1.0d0 + a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.65e+15) {
		tmp = z / b;
	} else if (y <= 2.4e-48) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.65e+15:
		tmp = z / b
	elif y <= 2.4e-48:
		tmp = x / (1.0 + a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.65e+15)
		tmp = Float64(z / b);
	elseif (y <= 2.4e-48)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.65e+15)
		tmp = z / b;
	elseif (y <= 2.4e-48)
		tmp = x / (1.0 + a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.65e+15], N[(z / b), $MachinePrecision], If[LessEqual[y, 2.4e-48], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e15 or 2.4e-48 < y

   1. Initial program 75.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   if -1.65e15 < y < 2.4e-48

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 41.8%

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 41.8%

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

Alternative 14: 42.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+125}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;a \leq 245000000000:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -3.3e+125)
   (/ x a)
   (if (<= a -1.6e-193)
     (/ z b)
     (if (<= a 2.4e-33)
       (/ x 1.0)
       (if (<= a 245000000000.0) (/ z b) (/ x a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.3e+125) {
		tmp = x / a;
	} else if (a <= -1.6e-193) {
		tmp = z / b;
	} else if (a <= 2.4e-33) {
		tmp = x / 1.0;
	} else if (a <= 245000000000.0) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	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 (a <= (-3.3d+125)) then
        tmp = x / a
    else if (a <= (-1.6d-193)) then
        tmp = z / b
    else if (a <= 2.4d-33) then
        tmp = x / 1.0d0
    else if (a <= 245000000000.0d0) then
        tmp = z / b
    else
        tmp = x / a
    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 (a <= -3.3e+125) {
		tmp = x / a;
	} else if (a <= -1.6e-193) {
		tmp = z / b;
	} else if (a <= 2.4e-33) {
		tmp = x / 1.0;
	} else if (a <= 245000000000.0) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -3.3e+125:
		tmp = x / a
	elif a <= -1.6e-193:
		tmp = z / b
	elif a <= 2.4e-33:
		tmp = x / 1.0
	elif a <= 245000000000.0:
		tmp = z / b
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -3.3e+125)
		tmp = Float64(x / a);
	elseif (a <= -1.6e-193)
		tmp = Float64(z / b);
	elseif (a <= 2.4e-33)
		tmp = Float64(x / 1.0);
	elseif (a <= 245000000000.0)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -3.3e+125)
		tmp = x / a;
	elseif (a <= -1.6e-193)
		tmp = z / b;
	elseif (a <= 2.4e-33)
		tmp = x / 1.0;
	elseif (a <= 245000000000.0)
		tmp = z / b;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -3.3e+125], N[(x / a), $MachinePrecision], If[LessEqual[a, -1.6e-193], N[(z / b), $MachinePrecision], If[LessEqual[a, 2.4e-33], N[(x / 1.0), $MachinePrecision], If[LessEqual[a, 245000000000.0], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.3000000000000001e125 or 2.45e11 < a

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 41.8%

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 25.5%

         \[\leadsto \frac{x}{a} \]

   7. Applied rewrites 25.5%

      \[\leadsto \frac{x}{\color{blue}{a}} \]

   if -3.3000000000000001e125 < a < -1.6e-193 or 2.4e-33 < a < 2.45e11

   1. Initial program 75.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.3%

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

   4. Applied rewrites 34.3%

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

   if -1.6e-193 < a < 2.4e-33

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 41.8%

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{x}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.5%

      \[\leadsto \frac{x}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 41.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a + 1 \leq -50000000:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a + 1 \leq 2:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ a 1.0) -50000000.0)
   (/ x a)
   (if (<= (+ a 1.0) 2.0) (/ x 1.0) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a + 1.0) <= -50000000.0) {
		tmp = x / a;
	} else if ((a + 1.0) <= 2.0) {
		tmp = x / 1.0;
	} else {
		tmp = x / a;
	}
	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 ((a + 1.0d0) <= (-50000000.0d0)) then
        tmp = x / a
    else if ((a + 1.0d0) <= 2.0d0) then
        tmp = x / 1.0d0
    else
        tmp = x / a
    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 ((a + 1.0) <= -50000000.0) {
		tmp = x / a;
	} else if ((a + 1.0) <= 2.0) {
		tmp = x / 1.0;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a + 1.0) <= -50000000.0:
		tmp = x / a
	elif (a + 1.0) <= 2.0:
		tmp = x / 1.0
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a + 1.0) <= -50000000.0)
		tmp = Float64(x / a);
	elseif (Float64(a + 1.0) <= 2.0)
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a + 1.0) <= -50000000.0)
		tmp = x / a;
	elseif ((a + 1.0) <= 2.0)
		tmp = x / 1.0;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a + 1.0), $MachinePrecision], -50000000.0], N[(x / a), $MachinePrecision], If[LessEqual[N[(a + 1.0), $MachinePrecision], 2.0], N[(x / 1.0), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 a #s(literal 1 binary64)) < -5e7 or 2 < (+.f64 a #s(literal 1 binary64))

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 41.8%

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 25.5%

         \[\leadsto \frac{x}{a} \]

   7. Applied rewrites 25.5%

      \[\leadsto \frac{x}{\color{blue}{a}} \]

   if -5e7 < (+.f64 a #s(literal 1 binary64)) < 2

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 41.8%

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{x}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.5%

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

Alternative 16: 25.5% accurate, 5.5× speedup

Math

\[\frac{x}{a} \]

FPCore

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

C

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

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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

2. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

   2. lower-+.f64 41.8%

      \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

4. Applied rewrites 41.8%

   \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

5. Taylor expanded in a around inf

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

   1. lower-/.f64 25.5%

      \[\leadsto \frac{x}{a} \]

7. Applied rewrites 25.5%

   \[\leadsto \frac{x}{\color{blue}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025206 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64
  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))