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

Percentage Accurate: 98.2% → 98.9%

Time: 10.7s

Alternatives: 14

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 10^{+214}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t - a}, t - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 1e+214) (fma t_1 y x) (fma (/ y (- t a)) (- t z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 1e+214) {
		tmp = fma(t_1, y, x);
	} else {
		tmp = fma((y / (t - a)), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 1e+214)
		tmp = fma(t_1, y, x);
	else
		tmp = fma(Float64(y / Float64(t - a)), Float64(t - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+214], N[(t$95$1 * y + x), $MachinePrecision], N[(N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 9.9999999999999995e213

   1. Initial program 99.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   if 9.9999999999999995e213 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 70.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{0 - \left(a - t\right)}}, \mathsf{neg}\left(\left(z - t\right)\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \mathsf{neg}\left(\left(z - t\right)\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \mathsf{neg}\left(\left(z - t\right)\right), x\right) \]

      11. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \mathsf{neg}\left(\left(z - t\right)\right), x\right) \]

      12. neg-sub0 N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{t - a}, \color{blue}{0 - \left(z - t\right)}, x\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{t - a}, 0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}, x\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{t - a}, 0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}, x\right) \]

      18. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{t - a}, \color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}, x\right) \]

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

Alternative 2: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_1 \cdot y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -1000000000000.0)
     (* t_1 y)
     (if (<= t_1 5e-5)
       (fma y (/ (- z t) a) x)
       (if (<= t_1 2e+72) (fma y (- 1.0 (/ z t)) x) (fma z (/ y a) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_1 * y;
	} else if (t_1 <= 5e-5) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 2e+72) {
		tmp = fma(y, (1.0 - (z / t)), x);
	} else {
		tmp = fma(z, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = Float64(t_1 * y);
	elseif (t_1 <= 5e-5)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 2e+72)
		tmp = fma(y, Float64(1.0 - Float64(z / t)), x);
	else
		tmp = fma(z, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000000.0], N[(t$95$1 * y), $MachinePrecision], If[LessEqual[t$95$1, 5e-5], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+72], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e12

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 58.8

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

   5. Simplified 58.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 71.0

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

   7. Applied egg-rr 71.0%

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

   if -1e12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000024e-5

   1. Initial program 98.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 98.1

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

   5. Simplified 98.1%

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

   if 5.00000000000000024e-5 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.99999999999999989e72

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. /-lowering-/.f64 95.2

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

   5. Simplified 95.2%

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

   if 1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 85.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 85.1

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

   4. Applied egg-rr 85.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 75.7

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

   7. Simplified 75.7%

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

Alternative 3: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+72}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma y (- 1.0 (/ z t)) x)))
   (if (<= t_1 -1000000000000.0)
     t_2
     (if (<= t_1 5e-5)
       (fma y (/ (- z t) a) x)
       (if (<= t_1 2e+72) t_2 (fma z (/ y a) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma(y, (1.0 - (z / t)), x);
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-5) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 2e+72) {
		tmp = t_2;
	} else {
		tmp = fma(z, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(y, Float64(1.0 - Float64(z / t)), x)
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-5)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 2e+72)
		tmp = t_2;
	else
		tmp = fma(z, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000000.0], t$95$2, If[LessEqual[t$95$1, 5e-5], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+72], t$95$2, N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e12 or 5.00000000000000024e-5 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.99999999999999989e72

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. /-lowering-/.f64 85.5

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

   5. Simplified 85.5%

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

   if -1e12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000024e-5

   1. Initial program 98.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 98.1

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

   5. Simplified 98.1%

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

   if 1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 85.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 85.1

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

   4. Applied egg-rr 85.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 75.7

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

   7. Simplified 75.7%

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

Alternative 4: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+72}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma y (- 1.0 (/ z t)) x)))
   (if (<= t_1 -1000000000000.0)
     t_2
     (if (<= t_1 5e-6)
       (- x (* y (/ t a)))
       (if (<= t_1 2e+72) t_2 (fma z (/ y a) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma(y, (1.0 - (z / t)), x);
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-6) {
		tmp = x - (y * (t / a));
	} else if (t_1 <= 2e+72) {
		tmp = t_2;
	} else {
		tmp = fma(z, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(y, Float64(1.0 - Float64(z / t)), x)
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-6)
		tmp = Float64(x - Float64(y * Float64(t / a)));
	elseif (t_1 <= 2e+72)
		tmp = t_2;
	else
		tmp = fma(z, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000000.0], t$95$2, If[LessEqual[t$95$1, 5e-6], N[(x - N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+72], t$95$2, N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e12 or 5.00000000000000041e-6 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.99999999999999989e72

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. --lowering--.f64 N/A

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

      17. /-lowering-/.f64 84.9

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

   5. Simplified 84.9%

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

   if -1e12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000041e-6

   1. Initial program 98.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 98.8

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

   4. Applied egg-rr 98.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 98.8

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 92.0

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

   9. Simplified 92.0%

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

   10. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 91.5

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

   12. Simplified 91.5%

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

   if 1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 85.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 85.1

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

   4. Applied egg-rr 85.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 75.7

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

   7. Simplified 75.7%

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

Alternative 5: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := z \cdot \frac{y}{a - t}\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 5000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (* z (/ y (- a t)))))
   (if (<= t_1 -1000000000000.0)
     t_2
     (if (<= t_1 5e-11)
       (- x (* y (/ t a)))
       (if (<= t_1 5000000000000.0) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = z * (y / (a - t));
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-11) {
		tmp = x - (y * (t / a));
	} else if (t_1 <= 5000000000000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (a - t)
    t_2 = z * (y / (a - t))
    if (t_1 <= (-1000000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 5d-11) then
        tmp = x - (y * (t / a))
    else if (t_1 <= 5000000000000.0d0) then
        tmp = y + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = z * (y / (a - t));
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-11) {
		tmp = x - (y * (t / a));
	} else if (t_1 <= 5000000000000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	t_2 = z * (y / (a - t))
	tmp = 0
	if t_1 <= -1000000000000.0:
		tmp = t_2
	elif t_1 <= 5e-11:
		tmp = x - (y * (t / a))
	elif t_1 <= 5000000000000.0:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(z * Float64(y / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-11)
		tmp = Float64(x - Float64(y * Float64(t / a)));
	elseif (t_1 <= 5000000000000.0)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	t_2 = z * (y / (a - t));
	tmp = 0.0;
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-11)
		tmp = x - (y * (t / a));
	elseif (t_1 <= 5000000000000.0)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000000.0], t$95$2, If[LessEqual[t$95$1, 5e-11], N[(x - N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5000000000000.0], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e12 or 5e12 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 93.8%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 61.9

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

   5. Simplified 61.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 69.5

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

   7. Applied egg-rr 69.5%

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

   if -1e12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000018e-11

   1. Initial program 98.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 98.8

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

   4. Applied egg-rr 98.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 98.8

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 93.0

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

   9. Simplified 93.0%

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

   10. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 92.4

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

   12. Simplified 92.4%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5e12

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 93.7

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

   5. Simplified 93.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1000000000000:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := z \cdot \frac{y}{a - t}\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (* z (/ y (- a t)))))
   (if (<= t_1 -1000000000000.0)
     t_2
     (if (<= t_1 5e-5)
       (fma z (/ y a) x)
       (if (<= t_1 5000000000000.0) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = z * (y / (a - t));
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-5) {
		tmp = fma(z, (y / a), x);
	} else if (t_1 <= 5000000000000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(z * Float64(y / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-5)
		tmp = fma(z, Float64(y / a), x);
	elseif (t_1 <= 5000000000000.0)
		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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000000.0], t$95$2, If[LessEqual[t$95$1, 5e-5], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5000000000000.0], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e12 or 5e12 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 93.8%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 61.9

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

   5. Simplified 61.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 69.5

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

   7. Applied egg-rr 69.5%

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

   if -1e12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000024e-5

   1. Initial program 98.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 98.9

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 78.2

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

   7. Simplified 78.2%

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

   if 5.00000000000000024e-5 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5e12

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 95.9

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

   5. Simplified 95.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1000000000000:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := z \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+72}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (* z (/ y a))))
   (if (<= t_1 -5e+144)
     t_2
     (if (<= t_1 5e-62) x (if (<= t_1 2e+72) (+ y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = z * (y / a);
	double tmp;
	if (t_1 <= -5e+144) {
		tmp = t_2;
	} else if (t_1 <= 5e-62) {
		tmp = x;
	} else if (t_1 <= 2e+72) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z - t) / (a - t)
    t_2 = z * (y / a)
    if (t_1 <= (-5d+144)) then
        tmp = t_2
    else if (t_1 <= 5d-62) then
        tmp = x
    else if (t_1 <= 2d+72) then
        tmp = y + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = z * (y / a);
	double tmp;
	if (t_1 <= -5e+144) {
		tmp = t_2;
	} else if (t_1 <= 5e-62) {
		tmp = x;
	} else if (t_1 <= 2e+72) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	t_2 = z * (y / a)
	tmp = 0
	if t_1 <= -5e+144:
		tmp = t_2
	elif t_1 <= 5e-62:
		tmp = x
	elif t_1 <= 2e+72:
		tmp = y + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(z * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -5e+144)
		tmp = t_2;
	elseif (t_1 <= 5e-62)
		tmp = x;
	elseif (t_1 <= 2e+72)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	t_2 = z * (y / a);
	tmp = 0.0;
	if (t_1 <= -5e+144)
		tmp = t_2;
	elseif (t_1 <= 5e-62)
		tmp = x;
	elseif (t_1 <= 2e+72)
		tmp = y + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+144], t$95$2, If[LessEqual[t$95$1, 5e-62], x, If[LessEqual[t$95$1, 2e+72], N[(y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -4.9999999999999999e144 or 1.99999999999999989e72 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 89.4%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 67.6

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

   5. Simplified 67.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 76.2

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

   7. Applied egg-rr 76.2%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 51.3%

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

   if -4.9999999999999999e144 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.0000000000000002e-62

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 63.9%

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

   if 5.0000000000000002e-62 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.99999999999999989e72

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 84.7

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

   5. Simplified 84.7%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -5 \cdot 10^{+144}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{+72}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t} \cdot y\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+133}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+59}:\\ \;\;\;\;x + y \cdot \frac{t}{t - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) (- a t)) y)))
   (if (<= t_1 -4e+133)
     (* (- z t) (/ y (- a t)))
     (if (<= t_1 5e+59) (+ x (* y (/ t (- t a)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / (a - t)) * y;
	double tmp;
	if (t_1 <= -4e+133) {
		tmp = (z - t) * (y / (a - t));
	} else if (t_1 <= 5e+59) {
		tmp = x + (y * (t / (t - a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((z - t) / (a - t)) * y
    if (t_1 <= (-4d+133)) then
        tmp = (z - t) * (y / (a - t))
    else if (t_1 <= 5d+59) then
        tmp = x + (y * (t / (t - a)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / (a - t)) * y;
	double tmp;
	if (t_1 <= -4e+133) {
		tmp = (z - t) * (y / (a - t));
	} else if (t_1 <= 5e+59) {
		tmp = x + (y * (t / (t - a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((z - t) / (a - t)) * y
	tmp = 0
	if t_1 <= -4e+133:
		tmp = (z - t) * (y / (a - t))
	elif t_1 <= 5e+59:
		tmp = x + (y * (t / (t - a)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) / Float64(a - t)) * y)
	tmp = 0.0
	if (t_1 <= -4e+133)
		tmp = Float64(Float64(z - t) * Float64(y / Float64(a - t)));
	elseif (t_1 <= 5e+59)
		tmp = Float64(x + Float64(y * Float64(t / Float64(t - a))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z - t) / (a - t)) * y;
	tmp = 0.0;
	if (t_1 <= -4e+133)
		tmp = (z - t) * (y / (a - t));
	elseif (t_1 <= 5e+59)
		tmp = x + (y * (t / (t - a)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+133], N[(N[(z - t), $MachinePrecision] * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+59], N[(x + N[(y * N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < -4.0000000000000001e133

   1. Initial program 90.3%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 67.2

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

   5. Simplified 67.2%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 92.8

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

   7. Applied egg-rr 92.8%

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

   if -4.0000000000000001e133 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < 4.9999999999999997e59

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 99.4

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

   4. Applied egg-rr 99.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 99.3

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 87.7

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

   9. Simplified 87.7%

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

   if 4.9999999999999997e59 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))

   1. Initial program 96.5%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 48.5

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

   5. Simplified 48.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 84.2

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

   7. Applied egg-rr 84.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \cdot y \leq -4 \cdot 10^{+133}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;x + y \cdot \frac{t}{t - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma z (/ y a) x)))
   (if (<= t_1 5e-5) t_2 (if (<= t_1 5000000000000.0) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma(z, (y / a), x);
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = t_2;
	} else if (t_1 <= 5000000000000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(z, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = t_2;
	elseif (t_1 <= 5000000000000.0)
		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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], t$95$2, If[LessEqual[t$95$1, 5000000000000.0], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000024e-5 or 5e12 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 96.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 96.3

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

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 68.7

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

   7. Simplified 68.7%

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

   if 5.00000000000000024e-5 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5e12

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 95.9

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

   5. Simplified 95.9%

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

Alternative 10: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5000000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma y (/ z a) x)))
   (if (<= t_1 5e-5) t_2 (if (<= t_1 5000000000000.0) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma(y, (z / a), x);
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = t_2;
	} else if (t_1 <= 5000000000000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(y, Float64(z / a), x)
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = t_2;
	elseif (t_1 <= 5000000000000.0)
		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[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], t$95$2, If[LessEqual[t$95$1, 5000000000000.0], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 5.00000000000000024e-5 or 5e12 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 96.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 67.0

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

   5. Simplified 67.0%

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

   if 5.00000000000000024e-5 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5e12

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 95.9

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

   5. Simplified 95.9%

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

Alternative 11: 54.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t} \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+251}:\\ \;\;\;\;y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) (- a t)) y)))
   (if (<= t_1 -5e+251) y (if (<= t_1 2e+150) x y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / (a - t)) * y;
	double tmp;
	if (t_1 <= -5e+251) {
		tmp = y;
	} else if (t_1 <= 2e+150) {
		tmp = x;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_1 = ((z - t) / (a - t)) * y
    if (t_1 <= (-5d+251)) then
        tmp = y
    else if (t_1 <= 2d+150) then
        tmp = x
    else
        tmp = y
    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) / (a - t)) * y;
	double tmp;
	if (t_1 <= -5e+251) {
		tmp = y;
	} else if (t_1 <= 2e+150) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((z - t) / (a - t)) * y
	tmp = 0
	if t_1 <= -5e+251:
		tmp = y
	elif t_1 <= 2e+150:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) / Float64(a - t)) * y)
	tmp = 0.0
	if (t_1 <= -5e+251)
		tmp = y;
	elseif (t_1 <= 2e+150)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z - t) / (a - t)) * y;
	tmp = 0.0;
	if (t_1 <= -5e+251)
		tmp = y;
	elseif (t_1 <= 2e+150)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+251], y, If[LessEqual[t$95$1, 2e+150], x, y]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < -5.0000000000000005e251 or 1.99999999999999996e150 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 59.4

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

   5. Simplified 59.4%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 25.1%

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

   if -5.0000000000000005e251 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < 1.99999999999999996e150

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 65.9%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \cdot y \leq -5 \cdot 10^{+251}:\\ \;\;\;\;y\\ \mathbf{elif}\;\frac{z - t}{a - t} \cdot y \leq 2 \cdot 10^{+150}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 3.05 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- a t)) 3.05e-59) x (+ y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= 3.05e-59) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((z - t) / (a - t)) <= 3.05d-59) then
        tmp = x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= 3.05e-59) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (a - t)) <= 3.05e-59:
		tmp = x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(a - t)) <= 3.05e-59)
		tmp = x;
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (a - t)) <= 3.05e-59)
		tmp = x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], 3.05e-59], x, N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 3.0499999999999998e-59

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 57.3%

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

   if 3.0499999999999998e-59 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 69.9

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

   5. Simplified 69.9%

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

Alternative 13: 95.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. +-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. frac-2neg N/A

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

   5. associate-/r/ N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{0 - \left(a - t\right)}}, \mathsf{neg}\left(\left(z - t\right)\right), x\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{y}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \mathsf{neg}\left(\left(z - t\right)\right), x\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{y}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \mathsf{neg}\left(\left(z - t\right)\right), x\right) \]

   11. associate--r+ N/A

       \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \mathsf{neg}\left(\left(z - t\right)\right), x\right) \]

   12. neg-sub0 N/A

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

   13. remove-double-neg N/A

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

   14. --lowering--.f64 N/A

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

   15. neg-sub0 N/A

       \[\leadsto \mathsf{fma}\left(\frac{y}{t - a}, \color{blue}{0 - \left(z - t\right)}, x\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{y}{t - a}, 0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}, x\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{y}{t - a}, 0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)}, x\right) \]

   18. associate--r+ N/A

       \[\leadsto \mathsf{fma}\left(\frac{y}{t - a}, \color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - z}, x\right) \]

   19. neg-sub0 N/A

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

   20. remove-double-neg N/A

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

   21. --lowering--.f64 96.6

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

4. Applied egg-rr 96.6%

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

Alternative 14: 51.0% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in x around inf

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

5. Simplified 51.2%

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

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (< y -8.508084860551241e-17)
     t_1
     (if (< y 2.894426862792089e-49)
       (+ x (* (* y (- z t)) (/ 1.0 (- a t))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * ((z - t) / (a - t)))
    if (y < (-8.508084860551241d-17)) then
        tmp = t_1
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) * (1.0d0 / (a - t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	tmp = 0
	if y < -8.508084860551241e-17:
		tmp = t_1
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) * Float64(1.0 / Float64(a - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -8.508084860551241e-17], t$95$1, If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8508084860551241/100000000000000000000000000000000) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t)))))))

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