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

Percentage Accurate: 98.2% → 98.2%

Time: 8.5s

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: 98.2% 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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z - t}{\frac{z - a}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 0.0) (+ (/ (- z t) (/ (- z a) y)) x) (+ (* y t_1) 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 <= 0.0) {
		tmp = ((z - t) / ((z - a) / y)) + x;
	} else {
		tmp = (y * t_1) + 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) :: t_1
    real(8) :: tmp
    t_1 = (z - t) / (z - a)
    if (t_1 <= 0.0d0) then
        tmp = ((z - t) / ((z - a) / y)) + x
    else
        tmp = (y * t_1) + x
    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 tmp;
	if (t_1 <= 0.0) {
		tmp = ((z - t) / ((z - a) / y)) + x;
	} else {
		tmp = (y * t_1) + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if t_1 <= 0.0:
		tmp = ((z - t) / ((z - a) / y)) + x
	else:
		tmp = (y * t_1) + 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 <= 0.0)
		tmp = Float64(Float64(Float64(z - t) / Float64(Float64(z - a) / y)) + x);
	else
		tmp = Float64(Float64(y * t_1) + x);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -0.0

   1. Initial program 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -0.0 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 0:\\ \;\;\;\;\frac{z - t}{\frac{z - a}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ (- t) z) y x)))
   (if (<= t_1 -4e+42)
     t_2
     (if (<= t_1 1e-7)
       (fma (- t z) (/ y a) x)
       (if (<= t_1 2.0)
         (+ y x)
         (if (<= t_1 5e+138) t_2 (/ (* y t) (- a z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((-t / z), y, x);
	double tmp;
	if (t_1 <= -4e+42) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 2.0) {
		tmp = y + x;
	} else if (t_1 <= 5e+138) {
		tmp = t_2;
	} else {
		tmp = (y * t) / (a - z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(Float64(-t) / z), y, x)
	tmp = 0.0
	if (t_1 <= -4e+42)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(y + x);
	elseif (t_1 <= 5e+138)
		tmp = t_2;
	else
		tmp = Float64(Float64(y * t) / Float64(a - z));
	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]}, Block[{t$95$2 = N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+42], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+138], t$95$2, N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.00000000000000018e42 or 2 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000016e138

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 73.7%

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

   if -4.00000000000000018e42 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 97.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 5.00000000000000016e138 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. neg-mul-1 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2:\\ \;\;\;\;y + x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ (- t) z) y x)))
   (if (<= t_1 -4e+42)
     t_2
     (if (<= t_1 1e-7)
       (fma (/ t a) y x)
       (if (<= t_1 2.0)
         (+ y x)
         (if (<= t_1 5e+138) t_2 (/ (* y t) (- a z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((-t / z), y, x);
	double tmp;
	if (t_1 <= -4e+42) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2.0) {
		tmp = y + x;
	} else if (t_1 <= 5e+138) {
		tmp = t_2;
	} else {
		tmp = (y * t) / (a - z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(Float64(-t) / z), y, x)
	tmp = 0.0
	if (t_1 <= -4e+42)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = Float64(y + x);
	elseif (t_1 <= 5e+138)
		tmp = t_2;
	else
		tmp = Float64(Float64(y * t) / Float64(a - z));
	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]}, Block[{t$95$2 = N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+42], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+138], t$95$2, N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.00000000000000018e42 or 2 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000016e138

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 73.7%

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

   if -4.00000000000000018e42 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 97.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 85.6

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

   5. Applied rewrites 85.6%

      \[\leadsto x + y \cdot \color{blue}{\frac{t}{a}} \]
   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 85.6

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

   7. Applied rewrites 85.6%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 5.00000000000000016e138 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. neg-mul-1 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2:\\ \;\;\;\;y + x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, z - t, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 -4e+42)
     (fma (/ y z) (- z t) x)
     (if (<= t_1 1e-7)
       (fma (- t z) (/ y a) x)
       (if (<= t_1 5e+138) (fma (/ (- z t) z) y x) (/ (* y t) (- a z)))))))

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+42) {
		tmp = fma((y / z), (z - t), x);
	} else if (t_1 <= 1e-7) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 5e+138) {
		tmp = fma(((z - t) / z), y, x);
	} else {
		tmp = (y * t) / (a - z);
	}
	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+42)
		tmp = fma(Float64(y / z), Float64(z - t), x);
	elseif (t_1 <= 1e-7)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 5e+138)
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	else
		tmp = Float64(Float64(y * t) / Float64(a - z));
	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, -4e+42], N[(N[(y / z), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+138], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.00000000000000018e42

   1. Initial program 87.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 61.7

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

   5. Applied rewrites 61.7%

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

   7. Applied rewrites 70.9%

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

   if -4.00000000000000018e42 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 97.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000016e138

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 5.00000000000000016e138 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. neg-mul-1 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -4 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, z - t, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{y}{z}, z - t, x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ y z) (- z t) x)))
   (if (<= t_1 -4e+42)
     t_2
     (if (<= t_1 1e-7)
       (fma (- t z) (/ y a) x)
       (if (<= t_1 2.0) (fma (/ z (- z a)) 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 = fma((y / z), (z - t), x);
	double tmp;
	if (t_1 <= -4e+42) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 2.0) {
		tmp = fma((z / (z - a)), 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 = fma(Float64(y / z), Float64(z - t), x)
	tmp = 0.0
	if (t_1 <= -4e+42)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+42], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.00000000000000018e42 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 76.1%

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

   if -4.00000000000000018e42 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 97.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{y}{z}, z - t, x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ y z) (- z t) x)))
   (if (<= t_1 -4e+42)
     t_2
     (if (<= t_1 1e-7)
       (fma (- t z) (/ y a) x)
       (if (<= t_1 1.0) (+ 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 = fma((y / z), (z - t), x);
	double tmp;
	if (t_1 <= -4e+42) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 1.0) {
		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 = fma(Float64(y / z), Float64(z - t), x)
	tmp = 0.0
	if (t_1 <= -4e+42)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 1.0)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+42], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.00000000000000018e42 or 1 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 76.3%

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

   if -4.00000000000000018e42 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 97.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 7: 81.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ (- t) z) y x)))
   (if (<= t_1 -4e+42)
     t_2
     (if (<= t_1 1e-7) (fma (/ t a) y x) (if (<= t_1 2.0) (+ 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 = fma((-t / z), y, x);
	double tmp;
	if (t_1 <= -4e+42) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 2.0) {
		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 = fma(Float64(Float64(-t) / z), y, x)
	tmp = 0.0
	if (t_1 <= -4e+42)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 2.0)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+42], t$95$2, If[LessEqual[t$95$1, 1e-7], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -4.00000000000000018e42 or 2 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 93.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 71.3%

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

   if -4.00000000000000018e42 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 97.6%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 85.6

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

   5. Applied rewrites 85.6%

      \[\leadsto x + y \cdot \color{blue}{\frac{t}{a}} \]
   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 85.6

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

   7. Applied rewrites 85.6%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 8: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+46}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\frac{-t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ y a) t x)))
   (if (<= t_1 1e-7)
     t_2
     (if (<= t_1 2e+46) (+ y x) (if (<= t_1 2e+188) (* (/ (- t) z) y) 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 = fma((y / a), t, x);
	double tmp;
	if (t_1 <= 1e-7) {
		tmp = t_2;
	} else if (t_1 <= 2e+46) {
		tmp = y + x;
	} else if (t_1 <= 2e+188) {
		tmp = (-t / z) * y;
	} 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 = fma(Float64(y / a), t, x)
	tmp = 0.0
	if (t_1 <= 1e-7)
		tmp = t_2;
	elseif (t_1 <= 2e+46)
		tmp = Float64(y + x);
	elseif (t_1 <= 2e+188)
		tmp = Float64(Float64(Float64(-t) / z) * y);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-7], t$95$2, If[LessEqual[t$95$1, 2e+46], N[(y + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+188], N[(N[((-t) / z), $MachinePrecision] * y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8 or 2e188 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.1%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e46

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 97.2

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

   5. Applied rewrites 97.2%

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

   if 2e46 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e188

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. neg-mul-1 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 70.1%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\frac{-t}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+46}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-7)
     (fma (/ y a) t x)
     (if (<= t_1 2e+46) (+ y x) (* (/ (- y) z) t)))))

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 <= 1e-7) {
		tmp = fma((y / a), t, x);
	} else if (t_1 <= 2e+46) {
		tmp = y + x;
	} else {
		tmp = (-y / z) * t;
	}
	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 <= 1e-7)
		tmp = fma(Float64(y / a), t, x);
	elseif (t_1 <= 2e+46)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(Float64(-y) / z) * t);
	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, 1e-7], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+46], N[(y + x), $MachinePrecision], N[(N[((-y) / z), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8

   1. Initial program 95.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 74.1

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

   5. Applied rewrites 74.1%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e46

   1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 97.2

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

   5. Applied rewrites 97.2%

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

   if 2e46 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 95.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 61.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{if}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+153}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ y a) t x)))
   (if (<= t_1 1e-7) t_2 (if (<= t_1 1e+153) (+ 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 = fma((y / a), t, x);
	double tmp;
	if (t_1 <= 1e-7) {
		tmp = t_2;
	} else if (t_1 <= 1e+153) {
		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 = fma(Float64(y / a), t, x)
	tmp = 0.0
	if (t_1 <= 1e-7)
		tmp = t_2;
	elseif (t_1 <= 1e+153)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-7], t$95$2, If[LessEqual[t$95$1, 1e+153], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999995e-8 or 1e153 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 94.4%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   if 9.9999999999999995e-8 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e153

   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 87.2

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

   5. Applied rewrites 87.2%

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

Alternative 11: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (<= t_1 (- INFINITY)) (+ (/ (* y (- z t)) (- z a)) x) (+ t_1 x))))

C

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

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((y * (z - t)) / (z - a)) + x;
	} else {
		tmp = t_1 + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (z - a))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((y * (z - t)) / (z - a)) + x
	else:
		tmp = t_1 + x
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * Float64(z - t)) / Float64(z - a)) + x);
	else
		tmp = Float64(t_1 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (z - a));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = ((y * (z - t)) / (z - a)) + x;
	else
		tmp = t_1 + x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -inf.0

   1. Initial program 64.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 98.7%

      \[x + y \cdot \frac{z - t}{z - a} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{z - a} \leq -\infty:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{z - a} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+246}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 2e+246) (+ y x) (* (/ y a) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 2e+246) {
		tmp = y + x;
	} else {
		tmp = (y / a) * t;
	}
	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)) <= 2d+246) then
        tmp = y + x
    else
        tmp = (y / a) * t
    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)) <= 2e+246) {
		tmp = y + x;
	} else {
		tmp = (y / a) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 2e+246:
		tmp = y + x
	else:
		tmp = (y / a) * t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 2e+246)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / a) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (z - a)) <= 2e+246)
		tmp = y + x;
	else
		tmp = (y / a) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 2e+246], N[(y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000014e246

   1. Initial program 97.5%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

   if 2.00000000000000014e246 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. neg-mul-1 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 92.4

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 69.3%

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

   12. Applied rewrites 69.3%

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

Alternative 13: 62.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+246}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 2e+246) (+ y x) (* (/ t a) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 2e+246) {
		tmp = y + x;
	} else {
		tmp = (t / a) * y;
	}
	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)) <= 2d+246) then
        tmp = y + x
    else
        tmp = (t / a) * y
    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)) <= 2e+246) {
		tmp = y + x;
	} else {
		tmp = (t / a) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 2e+246:
		tmp = y + x
	else:
		tmp = (t / a) * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 2e+246)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(t / a) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (z - a)) <= 2e+246)
		tmp = y + x;
	else
		tmp = (t / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 2e+246], N[(y + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 2.00000000000000014e246

   1. Initial program 97.5%

      \[x + y \cdot \frac{z - t}{z - a} \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

   if 2.00000000000000014e246 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. neg-mul-1 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 92.4

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 69.3%

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

   12. Applied rewrites 62.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+246}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Final simplification 96.9%

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

Alternative 15: 61.6% 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 96.9%

   \[x + y \cdot \frac{z - t}{z - a} \]
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 62.5

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

5. Applied rewrites 62.5%

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

Developer Target 1: 98.4% 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 2024277 
(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)))))