Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A

Percentage Accurate: 98.3% → 98.3%

Time: 7.0s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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 * ((z - t) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * ((z - t) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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 * ((z / (z - a)) - (t / (z - a))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 98.2

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

4. Applied rewrites 98.2%

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

Alternative 2: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z - a}\\ \mathbf{elif}\;t\_1 \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x + \mathsf{fma}\left(y, \frac{a - t}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+160)
     (/ (* (- t) y) (- z a))
     (if (<= t_1 -400000000000.0)
       (fma (/ (- t) z) y x)
       (if (<= t_1 0.001)
         (fma (/ (- t z) a) y x)
         (if (<= t_1 2.0) (+ x (fma y (/ (- a t) z) y)) (fma (/ t a) y x)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+160) {
		tmp = (-t * y) / (z - a);
	} else if (t_1 <= -400000000000.0) {
		tmp = fma((-t / z), y, x);
	} else if (t_1 <= 0.001) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = x + fma(y, ((a - t) / z), y);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+160)
		tmp = Float64(Float64(Float64(-t) * y) / Float64(z - a));
	elseif (t_1 <= -400000000000.0)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif (t_1 <= 0.001)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x + fma(y, Float64(Float64(a - t) / z), y));
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+160], N[(N[((-t) * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -400000000000.0], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 0.001], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x + N[(y * N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.0000000000000002e160

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 92.6%

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

   10. Applied rewrites 99.8%

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

   if -5.0000000000000002e160 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4e11

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 91.2

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

   7. Applied rewrites 91.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 91.2%

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

   if -4e11 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-3

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.3

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 98.4

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

   10. Applied rewrites 98.4%

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

   if 1e-3 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 42.1

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-/l* N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 99.0%

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

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.5

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

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{a}, y, x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2:\\ \;\;\;\;x + \mathsf{fma}\left(y, \frac{a - t}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z - a}\\ \mathbf{elif}\;t\_1 \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+160)
     (/ (* (- t) y) (- z a))
     (if (<= t_1 -400000000000.0)
       (fma (/ (- t) z) y x)
       (if (<= t_1 1e-89)
         (fma (/ (- t z) a) y x)
         (if (<= t_1 2.0) (fma (/ z (- z a)) y x) (fma (/ t a) y x)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+160) {
		tmp = (-t * y) / (z - a);
	} else if (t_1 <= -400000000000.0) {
		tmp = fma((-t / z), y, x);
	} else if (t_1 <= 1e-89) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+160)
		tmp = Float64(Float64(Float64(-t) * y) / Float64(z - a));
	elseif (t_1 <= -400000000000.0)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif (t_1 <= 1e-89)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+160], N[(N[((-t) * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -400000000000.0], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-89], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.0000000000000002e160

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 92.6%

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

   10. Applied rewrites 99.8%

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

   if -5.0000000000000002e160 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4e11

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 91.2

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

   7. Applied rewrites 91.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 91.2%

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

   if -4e11 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000004e-89

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 99.0

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

   10. Applied rewrites 99.0%

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

   if 1.00000000000000004e-89 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.5

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

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{a}, y, x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+160}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;t\_1 \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+160)
     (* (- t) (/ y (- z a)))
     (if (<= t_1 -400000000000.0)
       (fma (/ (- t) z) y x)
       (if (<= t_1 1e-89)
         (fma (/ (- t z) a) y x)
         (if (<= t_1 2.0) (fma (/ z (- z a)) y x) (fma (/ t a) y x)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+160) {
		tmp = -t * (y / (z - a));
	} else if (t_1 <= -400000000000.0) {
		tmp = fma((-t / z), y, x);
	} else if (t_1 <= 1e-89) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+160)
		tmp = Float64(Float64(-t) * Float64(y / Float64(z - a)));
	elseif (t_1 <= -400000000000.0)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif (t_1 <= 1e-89)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+160], N[((-t) * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -400000000000.0], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-89], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5.0000000000000002e160

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 92.6%

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

   if -5.0000000000000002e160 < (/.f64 (-.f64 z t) (-.f64 z a)) < -4e11

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 91.2

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

   7. Applied rewrites 91.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 91.2%

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

   if -4e11 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000004e-89

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 99.0

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

   10. Applied rewrites 99.0%

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

   if 1.00000000000000004e-89 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.5

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

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{a}, y, x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+160}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -400000000000.0)
     (fma (/ (- t) z) y x)
     (if (<= t_1 1e-89)
       (fma (/ (- t z) a) y x)
       (if (<= t_1 2.0) (fma (/ z (- z a)) y x) (fma (/ t a) y x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -400000000000.0) {
		tmp = fma((-t / z), y, x);
	} else if (t_1 <= 1e-89) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -400000000000.0)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif (t_1 <= 1e-89)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -400000000000.0], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-89], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4e11

   1. Initial program 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 72.8

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

   7. Applied rewrites 72.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.8%

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

   if -4e11 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.00000000000000004e-89

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 99.0

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

   10. Applied rewrites 99.0%

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

   if 1.00000000000000004e-89 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.5

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

   7. Applied rewrites 76.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-34} \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -400000000000.0)
     (fma (/ (- t) z) y x)
     (if (or (<= t_1 4e-34) (not (<= t_1 2.0))) (fma (/ t a) y x) (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -400000000000.0) {
		tmp = fma((-t / z), y, x);
	} else if ((t_1 <= 4e-34) || !(t_1 <= 2.0)) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -400000000000.0)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif ((t_1 <= 4e-34) || !(t_1 <= 2.0))
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -400000000000.0], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[Or[LessEqual[t$95$1, 4e-34], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4e11

   1. Initial program 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 72.8

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

   7. Applied rewrites 72.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.8%

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

   if -4e11 < (/.f64 (-.f64 z t) (-.f64 z a)) < 3.99999999999999971e-34 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 83.1

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

   5. Applied rewrites 83.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.1

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

   7. Applied rewrites 83.1%

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

   if 3.99999999999999971e-34 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 95.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -400000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 4 \cdot 10^{-34} \lor \neg \left(\frac{z - t}{z - a} \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -400000000000:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-34} \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -400000000000.0)
     (* (- t) (/ y z))
     (if (or (<= t_1 4e-34) (not (<= t_1 2.0))) (fma (/ t a) y x) (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -400000000000.0) {
		tmp = -t * (y / z);
	} else if ((t_1 <= 4e-34) || !(t_1 <= 2.0)) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -400000000000.0)
		tmp = Float64(Float64(-t) * Float64(y / z));
	elseif ((t_1 <= 4e-34) || !(t_1 <= 2.0))
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -400000000000.0], N[((-t) * N[(y / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 4e-34], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4e11

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 56.9%

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

   if -4e11 < (/.f64 (-.f64 z t) (-.f64 z a)) < 3.99999999999999971e-34 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 83.1

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

   5. Applied rewrites 83.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.1

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

   7. Applied rewrites 83.1%

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

   if 3.99999999999999971e-34 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 95.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -400000000000:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 4 \cdot 10^{-34} \lor \neg \left(\frac{z - t}{z - a} \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{t \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-34}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+100}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* t y) a)))
   (if (<= t_1 -1e+145)
     t_2
     (if (<= t_1 4e-34) (* 1.0 x) (if (<= t_1 4e+100) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (t * y) / a;
	double tmp;
	if (t_1 <= -1e+145) {
		tmp = t_2;
	} else if (t_1 <= 4e-34) {
		tmp = 1.0 * x;
	} else if (t_1 <= 4e+100) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    t_2 = (t * y) / a
    if (t_1 <= (-1d+145)) then
        tmp = t_2
    else if (t_1 <= 4d-34) then
        tmp = 1.0d0 * x
    else if (t_1 <= 4d+100) then
        tmp = y + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (t * y) / a;
	double tmp;
	if (t_1 <= -1e+145) {
		tmp = t_2;
	} else if (t_1 <= 4e-34) {
		tmp = 1.0 * x;
	} else if (t_1 <= 4e+100) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	t_2 = (t * y) / a
	tmp = 0
	if t_1 <= -1e+145:
		tmp = t_2
	elif t_1 <= 4e-34:
		tmp = 1.0 * x
	elif t_1 <= 4e+100:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(t * y) / a)
	tmp = 0.0
	if (t_1 <= -1e+145)
		tmp = t_2;
	elseif (t_1 <= 4e-34)
		tmp = Float64(1.0 * x);
	elseif (t_1 <= 4e+100)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	t_2 = (t * y) / a;
	tmp = 0.0;
	if (t_1 <= -1e+145)
		tmp = t_2;
	elseif (t_1 <= 4e-34)
		tmp = 1.0 * x;
	elseif (t_1 <= 4e+100)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+145], t$95$2, If[LessEqual[t$95$1, 4e-34], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$1, 4e+100], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -9.9999999999999999e144 or 4.00000000000000006e100 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.3%

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

   if -9.9999999999999999e144 < (/.f64 (-.f64 z t) (-.f64 z a)) < 3.99999999999999971e-34

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 94.8

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 67.1%

       \[\leadsto 1 \cdot x \]

   if 3.99999999999999971e-34 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.00000000000000006e100

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 85.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 85.6%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 4 \cdot 10^{-34}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 4 \cdot 10^{+100}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -5e+16)
     (fma (/ (- t) z) y x)
     (if (<= t_1 2.0) (fma (/ z (- z a)) y x) (fma (/ t a) y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= -5e+16) {
		tmp = fma((-t / z), y, x);
	} else if (t_1 <= 2.0) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -5e+16)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+16], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -5e16

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 72.0

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

   7. Applied rewrites 72.0%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.0%

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

   if -5e16 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 93.7

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

   5. Applied rewrites 93.7%

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

   if 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.5

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

   7. Applied rewrites 76.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-34} \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 4e-34) (not (<= t_1 2.0))) (fma (/ t a) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 4e-34) || !(t_1 <= 2.0)) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 4e-34) || !(t_1 <= 2.0))
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 4e-34], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 3.99999999999999971e-34 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.6

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

   7. Applied rewrites 74.6%

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

   if 3.99999999999999971e-34 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 95.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 4 \cdot 10^{-34} \lor \neg \left(\frac{z - t}{z - a} \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-34} \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 4e-34) (not (<= t_1 2.0))) (fma (/ y a) t x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 4e-34) || !(t_1 <= 2.0)) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 4e-34) || !(t_1 <= 2.0))
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 4e-34], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 3.99999999999999971e-34 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if 3.99999999999999971e-34 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 95.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 4 \cdot 10^{-34} \lor \neg \left(\frac{z - t}{z - a} \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+40} \lor \neg \left(z \leq 8.4 \cdot 10^{+33}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.8e+40) (not (<= z 8.4e+33)))
   (fma (- 1.0 (/ t z)) y x)
   (fma (/ t a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.8e+40) || !(z <= 8.4e+33)) {
		tmp = fma((1.0 - (t / z)), y, x);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.8e+40) || !(z <= 8.4e+33))
		tmp = fma(Float64(1.0 - Float64(t / z)), y, x);
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.8e+40], N[Not[LessEqual[z, 8.4e+33]], $MachinePrecision]], N[(N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8e40 or 8.4000000000000002e33 < z

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 90.7

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

   7. Applied rewrites 90.7%

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

   if -4.8e40 < z < 8.4000000000000002e33

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 81.3

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

   5. Applied rewrites 81.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.3

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

   7. Applied rewrites 81.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+40} \lor \neg \left(z \leq 8.4 \cdot 10^{+33}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 7.2 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 7.2e-16) (* 1.0 x) (+ y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 7.2e-16) {
		tmp = 1.0 * x;
	} else {
		tmp = y + x;
	}
	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) :: tmp
    if (((z - t) / (z - a)) <= 7.2d-16) then
        tmp = 1.0d0 * x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 7.2e-16) {
		tmp = 1.0 * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 7.2e-16:
		tmp = 1.0 * x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 7.2e-16)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (z - a)) <= 7.2e-16)
		tmp = 1.0 * x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 7.2e-16], N[(1.0 * x), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 7.19999999999999965e-16

   1. Initial program 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 89.9

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

   7. Applied rewrites 89.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 59.5%

       \[\leadsto 1 \cdot x \]

   if 7.19999999999999965e-16 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.4%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 76.3

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 76.3%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 7.2 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * ((z - t) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.2%

   \[x + y \cdot \frac{z - t}{z - a} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 15: 60.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ y + x \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return 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 = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{y + x} \]

   2. lower-+.f64 56.5

      \[\leadsto \color{blue}{y + x} \]

5. Applied rewrites 56.5%

   \[\leadsto \color{blue}{y + x} \]
6. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{\frac{z - a}{z - t}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024326 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ y (/ (- z a) (- z t)))))

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