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

Percentage Accurate: 74.1% → 90.8%

Time: 12.9s

Alternatives: 14

Speedup: 0.9×

Specification

Math

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

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: 74.1% accurate, 1.0× speedup

Math

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

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: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{\frac{y}{t} + \frac{x}{z}}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)} \cdot z\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{y}}{b}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+238}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (* (/ (+ (/ y t) (/ x z)) (fma (/ y t) b (+ 1.0 a))) z)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -2e-305)
       t_1
       (if (<= t_1 0.0)
         (/ (/ (fma t x (* z y)) y) b)
         (if (<= t_1 4e+238) t_1 (if (<= t_1 INFINITY) 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 = (((y / t) + (x / z)) / fma((y / t), b, (1.0 + a))) * z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -2e-305) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (fma(t, x, (z * y)) / y) / b;
	} else if (t_1 <= 4e+238) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		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(Float64(Float64(Float64(y / t) + Float64(x / z)) / fma(Float64(y / t), b, Float64(1.0 + a))) * z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -2e-305)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(t, x, Float64(z * y)) / y) / b);
	elseif (t_1 <= 4e+238)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	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[(N[(N[(N[(y / t), $MachinePrecision] + N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -2e-305], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 4e+238], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $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 or 4.0000000000000002e238 < (/.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 29.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 93.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99999999999999999e-305 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.0000000000000002e238

   1. Initial program 99.8%

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

   if -1.99999999999999999e-305 < (/.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 56.0%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 73.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 76.7%

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

   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 0.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

Alternative 2: 90.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{y}}{b}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (/ (fma (/ z t) y x) (fma (/ b t) y (- a -1.0)))
     (if (<= t_1 -2e-305)
       t_1
       (if (<= t_1 0.0)
         (/ (/ (fma t x (* z y)) y) b)
         (if (<= t_1 5e+275)
           t_1
           (if (<= t_1 INFINITY)
             (* y (/ z (fma b y (fma a t t))))
             (/ 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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((z / t), y, x) / fma((b / t), y, (a - -1.0));
	} else if (t_1 <= -2e-305) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (fma(t, x, (z * y)) / y) / b;
	} else if (t_1 <= 5e+275) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y * (z / fma(b, y, fma(a, t, t)));
	} 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)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(a - -1.0)));
	elseif (t_1 <= -2e-305)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(t, x, Float64(z * y)) / y) / b);
	elseif (t_1 <= 5e+275)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(y * Float64(z / fma(b, y, fma(a, t, t))));
	else
		tmp = Float64(z / b);
	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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-305], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 5e+275], t$95$1, If[LessEqual[t$95$1, Infinity], N[(y * N[(z / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 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 18.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. 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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 70.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 70.2

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

      17. lift-+.f64 N/A

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

      18. metadata-eval N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower--.f64 N/A

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

      24. metadata-eval 70.2

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

   4. Applied rewrites 70.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99999999999999999e-305 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5.0000000000000003e275

   1. Initial program 99.8%

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

   if -1.99999999999999999e-305 < (/.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 56.0%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 73.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 76.7%

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

   if 5.0000000000000003e275 < (/.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 23.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.9%

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

   10. Applied rewrites 93.0%

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

   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 0.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

Alternative 3: 74.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{y}}{b}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 -2e-305)
     (/ (fma (/ y t) z x) (+ 1.0 a))
     (if (<= t_2 0.0)
       (/ (/ (fma t x (* z y)) y) b)
       (if (<= t_2 5e+275)
         (/ t_1 (+ 1.0 a))
         (if (<= t_2 INFINITY) (* y (/ z (fma b y (fma a t t)))) (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -2e-305) {
		tmp = fma((y / t), z, x) / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = (fma(t, x, (z * y)) / y) / b;
	} else if (t_2 <= 5e+275) {
		tmp = t_1 / (1.0 + a);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = y * (z / fma(b, y, fma(a, t, t)));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= -2e-305)
		tmp = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(fma(t, x, Float64(z * y)) / y) / b);
	elseif (t_2 <= 5e+275)
		tmp = Float64(t_1 / Float64(1.0 + a));
	elseif (t_2 <= Inf)
		tmp = Float64(y * Float64(z / fma(b, y, fma(a, t, t))));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-305], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 5e+275], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(y * N[(z / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99999999999999999e-305

   1. Initial program 86.8%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 72.8%

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

   if -1.99999999999999999e-305 < (/.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 56.0%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 73.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 76.7%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5.0000000000000003e275

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 74.4%

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

   if 5.0000000000000003e275 < (/.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 23.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.9%

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

   10. Applied rewrites 93.0%

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

   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 0.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

Alternative 4: 74.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{y}}{b}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (/ y t) z x) (+ 1.0 a))))
   (if (<= t_1 -2e-305)
     t_2
     (if (<= t_1 0.0)
       (/ (/ (fma t x (* z y)) y) b)
       (if (<= t_1 5e+275)
         t_2
         (if (<= t_1 INFINITY) (* y (/ z (fma b y (fma a t t)))) (/ 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 = fma((y / t), z, x) / (1.0 + a);
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (fma(t, x, (z * y)) / y) / b;
	} else if (t_1 <= 5e+275) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y * (z / fma(b, y, fma(a, t, t)));
	} 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(fma(Float64(y / t), z, x) / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(t, x, Float64(z * y)) / y) / b);
	elseif (t_1 <= 5e+275)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(y * Float64(z / fma(b, y, fma(a, t, t))));
	else
		tmp = Float64(z / b);
	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[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-305], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 5e+275], t$95$2, If[LessEqual[t$95$1, Infinity], N[(y * N[(z / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $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))) < -1.99999999999999999e-305 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5.0000000000000003e275

   1. Initial program 93.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 73.6%

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

   if -1.99999999999999999e-305 < (/.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 56.0%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 73.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 76.7%

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

   if 5.0000000000000003e275 < (/.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 23.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.9%

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

   10. Applied rewrites 93.0%

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

   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 0.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

Alternative 5: 74.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{y}{t\_1} \cdot z\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b y (fma a t t)))
        (t_2 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (/ (fma (/ y t) z x) (+ 1.0 a))))
   (if (<= t_2 -2e-305)
     t_3
     (if (<= t_2 0.0)
       (* (/ y t_1) z)
       (if (<= t_2 5e+275)
         t_3
         (if (<= t_2 INFINITY) (* y (/ z t_1)) (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, y, fma(a, t, t));
	double t_2 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_3 = fma((y / t), z, x) / (1.0 + a);
	double tmp;
	if (t_2 <= -2e-305) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = (y / t_1) * z;
	} else if (t_2 <= 5e+275) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = y * (z / t_1);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, y, fma(a, t, t))
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a))
	tmp = 0.0
	if (t_2 <= -2e-305)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(y / t_1) * z);
	elseif (t_2 <= 5e+275)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(y * Float64(z / t_1));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-305], t$95$3, If[LessEqual[t$95$2, 0.0], N[(N[(y / t$95$1), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$2, 5e+275], t$95$3, If[LessEqual[t$95$2, Infinity], N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], N[(z / b), $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))) < -1.99999999999999999e-305 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5.0000000000000003e275

   1. Initial program 93.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 73.6%

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

   if -1.99999999999999999e-305 < (/.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 56.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.3%

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

   if 5.0000000000000003e275 < (/.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 23.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.9%

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

   10. Applied rewrites 93.0%

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

   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 0.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

Alternative 6: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-82} \lor \neg \left(t \leq 4.8 \cdot 10^{-113}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-x\right) \cdot t}{-b}}{y} + \frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -3.9e-82) (not (<= t 4.8e-113)))
   (/ (fma (/ z t) y x) (fma (/ b t) y (- a -1.0)))
   (+ (/ (/ (* (- x) t) (- b)) y) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3.9e-82) || !(t <= 4.8e-113)) {
		tmp = fma((z / t), y, x) / fma((b / t), y, (a - -1.0));
	} else {
		tmp = (((-x * t) / -b) / y) + (z / b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -3.9e-82) || !(t <= 4.8e-113))
		tmp = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(a - -1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-x) * t) / Float64(-b)) / y) + Float64(z / b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -3.9e-82], N[Not[LessEqual[t, 4.8e-113]], $MachinePrecision]], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[((-x) * t), $MachinePrecision] / (-b)), $MachinePrecision] / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.89999999999999973e-82 or 4.80000000000000024e-113 < t

   1. Initial program 82.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. 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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 91.4

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

      17. lift-+.f64 N/A

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

      18. metadata-eval N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower--.f64 N/A

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

      24. metadata-eval 91.4

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

   4. Applied rewrites 91.4%

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

   if -3.89999999999999973e-82 < t < 4.80000000000000024e-113

   1. Initial program 60.5%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 74.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-82} \lor \neg \left(t \leq 4.8 \cdot 10^{-113}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-x\right) \cdot t}{-b}}{y} + \frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-73} \lor \neg \left(t \leq 2.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-x\right) \cdot t}{-b}}{y} + \frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1e-73) (not (<= t 2.7e-6)))
   (/ (fma (/ y t) z x) (+ 1.0 a))
   (+ (/ (/ (* (- x) t) (- b)) y) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1e-73) || !(t <= 2.7e-6)) {
		tmp = fma((y / t), z, x) / (1.0 + a);
	} else {
		tmp = (((-x * t) / -b) / y) + (z / b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1e-73) || !(t <= 2.7e-6))
		tmp = Float64(fma(Float64(y / t), z, x) / Float64(1.0 + a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-x) * t) / Float64(-b)) / y) + Float64(z / b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1e-73], N[Not[LessEqual[t, 2.7e-6]], $MachinePrecision]], N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[((-x) * t), $MachinePrecision] / (-b)), $MachinePrecision] / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.99999999999999997e-74 or 2.69999999999999998e-6 < t

   1. Initial program 82.9%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 79.1%

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

   if -9.99999999999999997e-74 < t < 2.69999999999999998e-6

   1. Initial program 63.4%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 58.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-73} \lor \neg \left(t \leq 2.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-x\right) \cdot t}{-b}}{y} + \frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-250}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (fma (/ y t) b (+ 1.0 a)))))
   (if (<= t -5.5e-82)
     t_1
     (if (<= t 6e-250)
       (/ (fma t (/ x y) z) b)
       (if (<= t 8.8e+30) (* (/ y (fma b y (fma a t t))) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / fma((y / t), b, (1.0 + a));
	double tmp;
	if (t <= -5.5e-82) {
		tmp = t_1;
	} else if (t <= 6e-250) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t <= 8.8e+30) {
		tmp = (y / fma(b, y, fma(a, t, t))) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)))
	tmp = 0.0
	if (t <= -5.5e-82)
		tmp = t_1;
	elseif (t <= 6e-250)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t <= 8.8e+30)
		tmp = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.5e-82], t$95$1, If[LessEqual[t, 6e-250], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t, 8.8e+30], N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.4999999999999998e-82 or 8.7999999999999999e30 < t

   1. Initial program 81.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 73.5%

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

   if -5.4999999999999998e-82 < t < 6.00000000000000032e-250

   1. Initial program 65.4%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 71.5%

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

   if 6.00000000000000032e-250 < t < 8.7999999999999999e30

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.6%

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

Alternative 9: 59.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 + a}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-250}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{+32}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 a))))
   (if (<= t -9.5e-73)
     t_1
     (if (<= t 6e-250)
       (/ (fma t (/ x y) z) b)
       (if (<= t 5.7e+32) (* (/ y (fma b y (fma a t t))) z) 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 <= -9.5e-73) {
		tmp = t_1;
	} else if (t <= 6e-250) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t <= 5.7e+32) {
		tmp = (y / fma(b, y, fma(a, t, t))) * z;
	} 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 <= -9.5e-73)
		tmp = t_1;
	elseif (t <= 6e-250)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t <= 5.7e+32)
		tmp = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z);
	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, -9.5e-73], t$95$1, If[LessEqual[t, 6e-250], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t, 5.7e+32], N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.50000000000000005e-73 or 5.7e32 < t

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 63.2%

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

   if -9.50000000000000005e-73 < t < 6.00000000000000032e-250

   1. Initial program 65.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 70.7%

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

   if 6.00000000000000032e-250 < t < 5.7e32

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.6%

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

Alternative 10: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-73} \lor \neg \left(t \leq 3.8 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -9.5e-73) (not (<= t 3.8e+15)))
   (/ x (+ 1.0 a))
   (/ (fma t (/ x y) z) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -9.5e-73) || !(t <= 3.8e+15)) {
		tmp = x / (1.0 + a);
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -9.5e-73) || !(t <= 3.8e+15))
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -9.5e-73], N[Not[LessEqual[t, 3.8e+15]], $MachinePrecision]], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.50000000000000005e-73 or 3.8e15 < t

   1. Initial program 82.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 62.8%

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

   if -9.50000000000000005e-73 < t < 3.8e15

   1. Initial program 64.3%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 67.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-73} \lor \neg \left(t \leq 3.8 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-82} \lor \neg \left(t \leq 4.5 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -7.2e-82) (not (<= t 4.5e+14))) (/ x (+ 1.0 a)) (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -7.2e-82) || !(t <= 4.5e+14)) {
		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 ((t <= (-7.2d-82)) .or. (.not. (t <= 4.5d+14))) 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 ((t <= -7.2e-82) || !(t <= 4.5e+14)) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -7.2e-82) or not (t <= 4.5e+14):
		tmp = x / (1.0 + a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -7.2e-82) || !(t <= 4.5e+14))
		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 ((t <= -7.2e-82) || ~((t <= 4.5e+14)))
		tmp = x / (1.0 + a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -7.2e-82], N[Not[LessEqual[t, 4.5e+14]], $MachinePrecision]], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.19999999999999996e-82 or 4.5e14 < t

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 62.0%

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

   if -7.19999999999999996e-82 < t < 4.5e14

   1. Initial program 64.4%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 59.1%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-82} \lor \neg \left(t \leq 4.5 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-46} \lor \neg \left(t \leq 1.4 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -3e-46) (not (<= t 1.4e+152))) (/ x a) (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3e-46) || !(t <= 1.4e+152)) {
		tmp = x / 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 ((t <= (-3d-46)) .or. (.not. (t <= 1.4d+152))) then
        tmp = x / 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 ((t <= -3e-46) || !(t <= 1.4e+152)) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -3e-46) or not (t <= 1.4e+152):
		tmp = x / a
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -3e-46) || !(t <= 1.4e+152))
		tmp = Float64(x / 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 ((t <= -3e-46) || ~((t <= 1.4e+152)))
		tmp = x / a;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -3e-46], N[Not[LessEqual[t, 1.4e+152]], $MachinePrecision]], N[(x / a), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.99999999999999987e-46 or 1.4000000000000001e152 < t

   1. Initial program 83.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{x}{a} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.8%

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

   if -2.99999999999999987e-46 < t < 1.4000000000000001e152

   1. Initial program 67.3%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 52.7%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-46} \lor \neg \left(t \leq 1.4 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -4.5e+16) (/ x 1.0) (if (<= t 1.4e+152) (/ z b) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -4.5e+16) {
		tmp = x / 1.0;
	} else if (t <= 1.4e+152) {
		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 (t <= (-4.5d+16)) then
        tmp = x / 1.0d0
    else if (t <= 1.4d+152) 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 (t <= -4.5e+16) {
		tmp = x / 1.0;
	} else if (t <= 1.4e+152) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -4.5e+16:
		tmp = x / 1.0
	elif t <= 1.4e+152:
		tmp = z / b
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -4.5e+16)
		tmp = Float64(x / 1.0);
	elseif (t <= 1.4e+152)
		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 (t <= -4.5e+16)
		tmp = x / 1.0;
	elseif (t <= 1.4e+152)
		tmp = z / b;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -4.5e+16], N[(x / 1.0), $MachinePrecision], If[LessEqual[t, 1.4e+152], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.5e16

   1. Initial program 84.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 39.6%

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

   if -4.5e16 < t < 1.4000000000000001e152

   1. Initial program 68.1%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 51.0%

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

   if 1.4000000000000001e152 < t

   1. Initial program 82.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{x}{a} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.4%

      \[\leadsto \frac{x}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 24.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{a} \end{array} \]

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 73.8%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 53.8%

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

6. Taylor expanded in a around inf

   \[\leadsto \frac{x}{a} \]
7. Step-by-step derivation

8. Applied rewrites 25.8%

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

9. Final simplification 25.8%

   \[\leadsto \frac{x}{a} \]
10. Add Preprocessing

Developer Target 1: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))
   (if (< t -1.3659085366310088e-271)
     t_1
     (if (< t 3.036967103737246e-130) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 * ((x + ((y / t) * z)) * (1.0d0 / ((a + 1.0d0) + ((y / t) * b))))
    if (t < (-1.3659085366310088d-271)) then
        tmp = t_1
    else if (t < 3.036967103737246d-130) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))))
	tmp = 0
	if t < -1.3659085366310088e-271:
		tmp = t_1
	elif t < 3.036967103737246e-130:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 * Float64(Float64(x + Float64(Float64(y / t) * z)) * Float64(1.0 / Float64(Float64(a + 1.0) + Float64(Float64(y / t) * b)))))
	tmp = 0.0
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	tmp = 0.0;
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 * N[(N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.3659085366310088e-271], t$95$1, If[Less[t, 3.036967103737246e-130], N[(z / b), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2025019 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1707385670788761/12500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 1518483551868623/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))