Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Percentage Accurate: 67.7% → 90.4%

Time: 12.6s

Alternatives: 16

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{\frac{a - t}{z - t}}\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-258}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (- y x) (/ (- a t) (- z t)))))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -2e-258)
     t_1
     (if (<= t_2 0.0) (fma (- x y) (/ (- z a) t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - x) / ((a - t) / (z - t)));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -2e-258) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma((x - y), ((z - a) / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - x) / Float64(Float64(a - t) / Float64(z - t))))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -2e-258)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(x - y), Float64(Float64(z - a) / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - x), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-258], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -1.99999999999999991e-258 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if -1.99999999999999991e-258 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.2%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

Alternative 2: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-258}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z t) (- a t)) (- y x) x))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -2e-258)
     t_1
     (if (<= t_2 0.0) (fma (- x y) (/ (- z a) t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - t) / (a - t)), (y - x), x);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -2e-258) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma((x - y), ((z - a) / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -2e-258)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(x - y), Float64(Float64(z - a) / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-258], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -1.99999999999999991e-258 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 73.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 90.2

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

   4. Applied rewrites 90.2%

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

   if -1.99999999999999991e-258 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.2%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

Alternative 3: 28.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - x\right)\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{x \cdot z}{t}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- y x))))
   (if (<= t -1.7e+144)
     t_1
     (if (<= t -1.2e-126)
       (/ (* x z) t)
       (if (<= t 1.7e-158)
         (/ (* y z) a)
         (if (<= t 3.4e+75) (* x (/ z t)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y - x);
	double tmp;
	if (t <= -1.7e+144) {
		tmp = t_1;
	} else if (t <= -1.2e-126) {
		tmp = (x * z) / t;
	} else if (t <= 1.7e-158) {
		tmp = (y * z) / a;
	} else if (t <= 3.4e+75) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = x + (y - x)
    if (t <= (-1.7d+144)) then
        tmp = t_1
    else if (t <= (-1.2d-126)) then
        tmp = (x * z) / t
    else if (t <= 1.7d-158) then
        tmp = (y * z) / a
    else if (t <= 3.4d+75) then
        tmp = x * (z / t)
    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 t_1 = x + (y - x);
	double tmp;
	if (t <= -1.7e+144) {
		tmp = t_1;
	} else if (t <= -1.2e-126) {
		tmp = (x * z) / t;
	} else if (t <= 1.7e-158) {
		tmp = (y * z) / a;
	} else if (t <= 3.4e+75) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y - x)
	tmp = 0
	if t <= -1.7e+144:
		tmp = t_1
	elif t <= -1.2e-126:
		tmp = (x * z) / t
	elif t <= 1.7e-158:
		tmp = (y * z) / a
	elif t <= 3.4e+75:
		tmp = x * (z / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y - x))
	tmp = 0.0
	if (t <= -1.7e+144)
		tmp = t_1;
	elseif (t <= -1.2e-126)
		tmp = Float64(Float64(x * z) / t);
	elseif (t <= 1.7e-158)
		tmp = Float64(Float64(y * z) / a);
	elseif (t <= 3.4e+75)
		tmp = Float64(x * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y - x);
	tmp = 0.0;
	if (t <= -1.7e+144)
		tmp = t_1;
	elseif (t <= -1.2e-126)
		tmp = (x * z) / t;
	elseif (t <= 1.7e-158)
		tmp = (y * z) / a;
	elseif (t <= 3.4e+75)
		tmp = x * (z / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.7e+144], t$95$1, If[LessEqual[t, -1.2e-126], N[(N[(x * z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 1.7e-158], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 3.4e+75], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.7e144 or 3.40000000000000011e75 < t

   1. Initial program 28.5%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 49.4

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

   5. Applied rewrites 49.4%

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

   if -1.7e144 < t < -1.20000000000000003e-126

   1. Initial program 74.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 48.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 22.4%

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

   if -1.20000000000000003e-126 < t < 1.7e-158

   1. Initial program 97.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 79.9

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

   4. Applied rewrites 79.9%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 44.2

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

   7. Applied rewrites 44.2%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 40.4%

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

   if 1.7e-158 < t < 3.40000000000000011e75

   1. Initial program 73.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 28.5%

       \[\leadsto \frac{x \cdot z}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 30.4%

       \[\leadsto \frac{z}{t} \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;x + \left(y - x\right)\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{x \cdot z}{t}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 29.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - x\right)\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x \cdot z}{t}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- y x))))
   (if (<= t -1.7e+144)
     t_1
     (if (<= t -1.6e-124)
       (/ (* x z) t)
       (if (<= t 1.7e-158)
         (* y (/ z a))
         (if (<= t 3.4e+75) (* x (/ z t)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y - x);
	double tmp;
	if (t <= -1.7e+144) {
		tmp = t_1;
	} else if (t <= -1.6e-124) {
		tmp = (x * z) / t;
	} else if (t <= 1.7e-158) {
		tmp = y * (z / a);
	} else if (t <= 3.4e+75) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = x + (y - x)
    if (t <= (-1.7d+144)) then
        tmp = t_1
    else if (t <= (-1.6d-124)) then
        tmp = (x * z) / t
    else if (t <= 1.7d-158) then
        tmp = y * (z / a)
    else if (t <= 3.4d+75) then
        tmp = x * (z / t)
    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 t_1 = x + (y - x);
	double tmp;
	if (t <= -1.7e+144) {
		tmp = t_1;
	} else if (t <= -1.6e-124) {
		tmp = (x * z) / t;
	} else if (t <= 1.7e-158) {
		tmp = y * (z / a);
	} else if (t <= 3.4e+75) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y - x)
	tmp = 0
	if t <= -1.7e+144:
		tmp = t_1
	elif t <= -1.6e-124:
		tmp = (x * z) / t
	elif t <= 1.7e-158:
		tmp = y * (z / a)
	elif t <= 3.4e+75:
		tmp = x * (z / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y - x))
	tmp = 0.0
	if (t <= -1.7e+144)
		tmp = t_1;
	elseif (t <= -1.6e-124)
		tmp = Float64(Float64(x * z) / t);
	elseif (t <= 1.7e-158)
		tmp = Float64(y * Float64(z / a));
	elseif (t <= 3.4e+75)
		tmp = Float64(x * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y - x);
	tmp = 0.0;
	if (t <= -1.7e+144)
		tmp = t_1;
	elseif (t <= -1.6e-124)
		tmp = (x * z) / t;
	elseif (t <= 1.7e-158)
		tmp = y * (z / a);
	elseif (t <= 3.4e+75)
		tmp = x * (z / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.7e+144], t$95$1, If[LessEqual[t, -1.6e-124], N[(N[(x * z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 1.7e-158], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e+75], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.7e144 or 3.40000000000000011e75 < t

   1. Initial program 28.5%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 49.4

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

   5. Applied rewrites 49.4%

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

   if -1.7e144 < t < -1.60000000000000002e-124

   1. Initial program 74.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 48.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 22.4%

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

   if -1.60000000000000002e-124 < t < 1.7e-158

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in a around inf

      \[\leadsto y \cdot \frac{z}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 40.3%

       \[\leadsto y \cdot \frac{z}{a} \]

   if 1.7e-158 < t < 3.40000000000000011e75

   1. Initial program 73.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 28.5%

       \[\leadsto \frac{x \cdot z}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 30.4%

       \[\leadsto \frac{z}{t} \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;x + \left(y - x\right)\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x \cdot z}{t}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{if}\;t \leq -66000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -6.4 \cdot 10^{-229}:\\ \;\;\;\;x - \frac{x \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- x y) t) y)))
   (if (<= t -66000.0)
     t_1
     (if (<= t -6.4e-229)
       (- x (/ (* x z) a))
       (if (<= t 1.2e-115) (/ (* (- y x) z) a) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((x - y) / t), y);
	double tmp;
	if (t <= -66000.0) {
		tmp = t_1;
	} else if (t <= -6.4e-229) {
		tmp = x - ((x * z) / a);
	} else if (t <= 1.2e-115) {
		tmp = ((y - x) * z) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(x - y) / t), y)
	tmp = 0.0
	if (t <= -66000.0)
		tmp = t_1;
	elseif (t <= -6.4e-229)
		tmp = Float64(x - Float64(Float64(x * z) / a));
	elseif (t <= 1.2e-115)
		tmp = Float64(Float64(Float64(y - x) * z) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -66000.0], t$95$1, If[LessEqual[t, -6.4e-229], N[(x - N[(N[(x * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e-115], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -66000 or 1.20000000000000011e-115 < t

   1. Initial program 46.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.0%

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

   if -66000 < t < -6.4000000000000003e-229

   1. Initial program 89.3%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.6%

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

   if -6.4000000000000003e-229 < t < 1.20000000000000011e-115

   1. Initial program 98.4%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 65.4%

      \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -66000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{elif}\;t \leq -6.4 \cdot 10^{-229}:\\ \;\;\;\;x - \frac{x \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x}{t}, y\right)\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -6.4 \cdot 10^{-229}:\\ \;\;\;\;x - \frac{x \cdot z}{a}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ x t) y)))
   (if (<= t -6.6e+36)
     t_1
     (if (<= t -6.4e-229)
       (- x (/ (* x z) a))
       (if (<= t 7.2e-142) (/ (* (- y x) z) a) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (x / t), y);
	double tmp;
	if (t <= -6.6e+36) {
		tmp = t_1;
	} else if (t <= -6.4e-229) {
		tmp = x - ((x * z) / a);
	} else if (t <= 7.2e-142) {
		tmp = ((y - x) * z) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(x / t), y)
	tmp = 0.0
	if (t <= -6.6e+36)
		tmp = t_1;
	elseif (t <= -6.4e-229)
		tmp = Float64(x - Float64(Float64(x * z) / a));
	elseif (t <= 7.2e-142)
		tmp = Float64(Float64(Float64(y - x) * z) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(x / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -6.6e+36], t$95$1, If[LessEqual[t, -6.4e-229], N[(x - N[(N[(x * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.2e-142], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.5999999999999997e36 or 7.20000000000000001e-142 < t

   1. Initial program 47.0%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 61.8%

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

   if -6.5999999999999997e36 < t < -6.4000000000000003e-229

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.4%

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

   if -6.4000000000000003e-229 < t < 7.20000000000000001e-142

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 70.1%

      \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t}, y\right)\\ \mathbf{elif}\;t \leq -6.4 \cdot 10^{-229}:\\ \;\;\;\;x - \frac{x \cdot z}{a}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.6e+34)
     t_1
     (if (<= t 1.4e-44) (fma (- z t) (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -1.6e+34) {
		tmp = t_1;
	} else if (t <= 1.4e-44) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -1.6e+34)
		tmp = t_1;
	elseif (t <= 1.4e-44)
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.6e+34], t$95$1, If[LessEqual[t, 1.4e-44], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5999999999999999e34 or 1.4e-44 < t

   1. Initial program 40.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 79.8%

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

   if -1.5999999999999999e34 < t < 1.4e-44

   1. Initial program 93.2%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 80.3

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

   5. Applied rewrites 80.3%

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

Alternative 8: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.6e+34) t_1 (if (<= t 1.4e-44) (fma z (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -1.6e+34) {
		tmp = t_1;
	} else if (t <= 1.4e-44) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -1.6e+34)
		tmp = t_1;
	elseif (t <= 1.4e-44)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.6e+34], t$95$1, If[LessEqual[t, 1.4e-44], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5999999999999999e34 or 1.4e-44 < t

   1. Initial program 40.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 79.8%

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

   if -1.5999999999999999e34 < t < 1.4e-44

   1. Initial program 93.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.9

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

   5. Applied rewrites 77.9%

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

Alternative 9: 69.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- x y) t) y)))
   (if (<= t -1.6e+34) t_1 (if (<= t 3.6e-44) (fma z (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((x - y) / t), y);
	double tmp;
	if (t <= -1.6e+34) {
		tmp = t_1;
	} else if (t <= 3.6e-44) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(x - y) / t), y)
	tmp = 0.0
	if (t <= -1.6e+34)
		tmp = t_1;
	elseif (t <= 3.6e-44)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.6e+34], t$95$1, If[LessEqual[t, 3.6e-44], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5999999999999999e34 or 3.5999999999999999e-44 < t

   1. Initial program 40.6%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.8%

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

   if -1.5999999999999999e34 < t < 3.5999999999999999e-44

   1. Initial program 93.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.9

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

   5. Applied rewrites 77.9%

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

Alternative 10: 51.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x}{t}, y\right)\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-110}:\\ \;\;\;\;x - \frac{x \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ x t) y)))
   (if (<= t -6.6e+36) t_1 (if (<= t 2.2e-110) (- x (/ (* x z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (x / t), y);
	double tmp;
	if (t <= -6.6e+36) {
		tmp = t_1;
	} else if (t <= 2.2e-110) {
		tmp = x - ((x * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(x / t), y)
	tmp = 0.0
	if (t <= -6.6e+36)
		tmp = t_1;
	elseif (t <= 2.2e-110)
		tmp = Float64(x - Float64(Float64(x * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(x / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -6.6e+36], t$95$1, If[LessEqual[t, 2.2e-110], N[(x - N[(N[(x * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5999999999999997e36 or 2.1999999999999999e-110 < t

   1. Initial program 43.5%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 62.8%

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

   if -6.5999999999999997e36 < t < 2.1999999999999999e-110

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 54.6%

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

Alternative 11: 51.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{x}{a}, x\right)\\ \mathbf{if}\;a \leq -7200:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ x a) x)))
   (if (<= a -7200.0) t_1 (if (<= a 6.8e+19) (fma z (/ x t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, (x / a), x);
	double tmp;
	if (a <= -7200.0) {
		tmp = t_1;
	} else if (a <= 6.8e+19) {
		tmp = fma(z, (x / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(x / a), x)
	tmp = 0.0
	if (a <= -7200.0)
		tmp = t_1;
	elseif (a <= 6.8e+19)
		tmp = fma(z, Float64(x / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(x / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -7200.0], t$95$1, If[LessEqual[a, 6.8e+19], N[(z * N[(x / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7200 or 6.8e19 < a

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 47.4%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 47.9%

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

   if -7200 < a < 6.8e19

   1. Initial program 64.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 57.2%

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

Alternative 12: 43.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x}{t}, y\right)\\ \mathbf{if}\;t \leq -7 \cdot 10^{-127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ x t) y)))
   (if (<= t -7e-127) t_1 (if (<= t 5.5e-142) (/ (* y z) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (x / t), y);
	double tmp;
	if (t <= -7e-127) {
		tmp = t_1;
	} else if (t <= 5.5e-142) {
		tmp = (y * z) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(x / t), y)
	tmp = 0.0
	if (t <= -7e-127)
		tmp = t_1;
	elseif (t <= 5.5e-142)
		tmp = Float64(Float64(y * z) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(x / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -7e-127], t$95$1, If[LessEqual[t, 5.5e-142], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.99999999999999979e-127 or 5.50000000000000023e-142 < t

   1. Initial program 54.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 55.9%

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

   if -6.99999999999999979e-127 < t < 5.50000000000000023e-142

   1. Initial program 96.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sqr-neg N/A

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

      10. difference-of-squares N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 78.7

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

   4. Applied rewrites 78.7%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 44.0

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

   7. Applied rewrites 44.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 39.3%

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

Alternative 13: 24.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - x\right)\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- y x))))
   (if (<= t -1.7e+144) t_1 (if (<= t 3.4e+75) (* x (/ z t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y - x);
	double tmp;
	if (t <= -1.7e+144) {
		tmp = t_1;
	} else if (t <= 3.4e+75) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = x + (y - x)
    if (t <= (-1.7d+144)) then
        tmp = t_1
    else if (t <= 3.4d+75) then
        tmp = x * (z / t)
    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 t_1 = x + (y - x);
	double tmp;
	if (t <= -1.7e+144) {
		tmp = t_1;
	} else if (t <= 3.4e+75) {
		tmp = x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y - x)
	tmp = 0
	if t <= -1.7e+144:
		tmp = t_1
	elif t <= 3.4e+75:
		tmp = x * (z / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y - x))
	tmp = 0.0
	if (t <= -1.7e+144)
		tmp = t_1;
	elseif (t <= 3.4e+75)
		tmp = Float64(x * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y - x);
	tmp = 0.0;
	if (t <= -1.7e+144)
		tmp = t_1;
	elseif (t <= 3.4e+75)
		tmp = x * (z / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.7e+144], t$95$1, If[LessEqual[t, 3.4e+75], N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7e144 or 3.40000000000000011e75 < t

   1. Initial program 28.5%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 49.4

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

   5. Applied rewrites 49.4%

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

   if -1.7e144 < t < 3.40000000000000011e75

   1. Initial program 83.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 21.2%

       \[\leadsto \frac{x \cdot z}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.1%

       \[\leadsto \frac{z}{t} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;x + \left(y - x\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 24.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - x\right)\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;z \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- y x))))
   (if (<= y -3.6e+71) t_1 (if (<= y 6.8e-66) (* z (/ x t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y - x);
	double tmp;
	if (y <= -3.6e+71) {
		tmp = t_1;
	} else if (y <= 6.8e-66) {
		tmp = z * (x / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = x + (y - x)
    if (y <= (-3.6d+71)) then
        tmp = t_1
    else if (y <= 6.8d-66) then
        tmp = z * (x / t)
    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 t_1 = x + (y - x);
	double tmp;
	if (y <= -3.6e+71) {
		tmp = t_1;
	} else if (y <= 6.8e-66) {
		tmp = z * (x / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y - x)
	tmp = 0
	if y <= -3.6e+71:
		tmp = t_1
	elif y <= 6.8e-66:
		tmp = z * (x / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y - x))
	tmp = 0.0
	if (y <= -3.6e+71)
		tmp = t_1;
	elseif (y <= 6.8e-66)
		tmp = Float64(z * Float64(x / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y - x);
	tmp = 0.0;
	if (y <= -3.6e+71)
		tmp = t_1;
	elseif (y <= 6.8e-66)
		tmp = z * (x / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.6e+71], t$95$1, If[LessEqual[y, 6.8e-66], N[(z * N[(x / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6e71 or 6.79999999999999994e-66 < y

   1. Initial program 67.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 31.7

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

   5. Applied rewrites 31.7%

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

   if -3.6e71 < y < 6.79999999999999994e-66

   1. Initial program 67.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 24.3%

       \[\leadsto \frac{x \cdot z}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 27.2%

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

Alternative 15: 18.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ x + \left(y - x\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x + (y - x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(y - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

3. Taylor expanded in t around inf

   \[\leadsto x + \color{blue}{\left(y - x\right)} \]
4. Step-by-step derivation

   1. lower--.f64 20.6

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

5. Applied rewrites 20.6%

   \[\leadsto x + \color{blue}{\left(y - x\right)} \]
6. Add Preprocessing

Alternative 16: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 67.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. mul-1-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

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

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

   10. *-lft-identity N/A

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

   11. lower-fma.f64 N/A

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

   12. lower--.f64 N/A

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

   13. distribute-neg-frac2 N/A

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

   14. lower-/.f64 N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

   17. distribute-neg-in N/A

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

   18. remove-double-neg N/A

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

   19. lower-+.f64 N/A

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

   20. lower-neg.f64 42.0

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

5. Applied rewrites 42.0%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 2.8%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer Target 1: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    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 t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024225 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))