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

Percentage Accurate: 85.3% → 98.4%

Time: 10.7s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

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(Float64(y * 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[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

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(Float64(y * 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[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

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

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

   7. --lowering--.f64 99.2

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

4. Applied egg-rr 99.2%

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

Alternative 2: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.06e-106)
   (fma (/ (- t z) t) y x)
   (if (<= t 2.9e+24) (fma (- z t) (/ y a) x) (fma y (- 1.0 (/ z t)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.06e-106) {
		tmp = fma(((t - z) / t), y, x);
	} else if (t <= 2.9e+24) {
		tmp = fma((z - t), (y / a), x);
	} else {
		tmp = fma(y, (1.0 - (z / t)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.06e-106)
		tmp = fma(Float64(Float64(t - z) / t), y, x);
	elseif (t <= 2.9e+24)
		tmp = fma(Float64(z - t), Float64(y / a), x);
	else
		tmp = fma(y, Float64(1.0 - Float64(z / t)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.06e-106], N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 2.9e+24], N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.06e-106

   1. Initial program 75.4%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 99.0

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 85.8

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

   7. Simplified 85.8%

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

   if -1.06e-106 < t < 2.89999999999999979e24

   1. Initial program 92.6%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 81.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 84.8

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

   7. Applied egg-rr 84.8%

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

   if 2.89999999999999979e24 < t

   1. Initial program 69.7%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \left(\frac{z}{t} - \color{blue}{1}\right), x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - \frac{z}{t}\right) + 1}, x\right) \]

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 90.0

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

   5. Simplified 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z t)) x)))
   (if (<= t -9.5e-107) t_1 (if (<= t 2e+24) (fma (- z t) (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (z / t)), x);
	double tmp;
	if (t <= -9.5e-107) {
		tmp = t_1;
	} else if (t <= 2e+24) {
		tmp = fma((z - t), (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(z / t)), x)
	tmp = 0.0
	if (t <= -9.5e-107)
		tmp = t_1;
	elseif (t <= 2e+24)
		tmp = fma(Float64(z - t), Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -9.5e-107], t$95$1, If[LessEqual[t, 2e+24], N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.4999999999999999e-107 or 2e24 < t

   1. Initial program 73.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \left(\frac{z}{t} - \color{blue}{1}\right), x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - \frac{z}{t}\right) + 1}, x\right) \]

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 87.6

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

   5. Simplified 87.6%

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

   if -9.4999999999999999e-107 < t < 2e24

   1. Initial program 92.6%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 81.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 84.8

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

   7. Applied egg-rr 84.8%

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

Alternative 4: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{if}\;t \leq -1.08 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z t)) x)))
   (if (<= t -1.08e-106) t_1 (if (<= t 6e+23) (fma y (/ (- z t) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (z / t)), x);
	double tmp;
	if (t <= -1.08e-106) {
		tmp = t_1;
	} else if (t <= 6e+23) {
		tmp = fma(y, ((z - t) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(z / t)), x)
	tmp = 0.0
	if (t <= -1.08e-106)
		tmp = t_1;
	elseif (t <= 6e+23)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.08e-106], t$95$1, If[LessEqual[t, 6e+23], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.08e-106 or 6.0000000000000002e23 < t

   1. Initial program 73.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \left(\frac{z}{t} - \color{blue}{1}\right), x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - \frac{z}{t}\right) + 1}, x\right) \]

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 87.6

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

   5. Simplified 87.6%

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

   if -1.08e-106 < t < 6.0000000000000002e23

   1. Initial program 92.6%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 84.7

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

   5. Simplified 84.7%

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

Alternative 5: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{if}\;t \leq -7 \cdot 10^{-125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.34 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z t)) x)))
   (if (<= t -7e-125) t_1 (if (<= t 1.34e+26) (fma z (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (z / t)), x);
	double tmp;
	if (t <= -7e-125) {
		tmp = t_1;
	} else if (t <= 1.34e+26) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(z / t)), x)
	tmp = 0.0
	if (t <= -7e-125)
		tmp = t_1;
	elseif (t <= 1.34e+26)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -7e-125], t$95$1, If[LessEqual[t, 1.34e+26], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.99999999999999995e-125 or 1.34000000000000007e26 < t

   1. Initial program 74.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 0 - \left(\frac{z}{t} - \color{blue}{1}\right), x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0 - \frac{z}{t}\right) + 1}, x\right) \]

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 86.2

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

   5. Simplified 86.2%

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

   if -6.99999999999999995e-125 < t < 1.34000000000000007e26

   1. Initial program 92.1%

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

   3. Taylor expanded in t around 0

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

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

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

      2. *-lowering-*.f64 77.5

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

   5. Simplified 77.5%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{a}} \]
   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-/l* 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 82.2

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

   7. Applied egg-rr 82.2%

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

Alternative 6: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-118}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 6500000000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(a, \frac{y}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.6e-118)
   (+ x y)
   (if (<= t 6500000000000.0) (fma z (/ y a) x) (+ x (fma a (/ y t) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.6e-118) {
		tmp = x + y;
	} else if (t <= 6500000000000.0) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = x + fma(a, (y / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.6e-118)
		tmp = Float64(x + y);
	elseif (t <= 6500000000000.0)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = Float64(x + fma(a, Float64(y / t), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.6e-118], N[(x + y), $MachinePrecision], If[LessEqual[t, 6500000000000.0], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(a * N[(y / t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.60000000000000002e-118

   1. Initial program 76.9%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 70.3

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

   5. Simplified 70.3%

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

   if -1.60000000000000002e-118 < t < 6.5e12

   1. Initial program 91.9%

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

   3. Taylor expanded in t around 0

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

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

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

      2. *-lowering-*.f64 77.8

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

   5. Simplified 77.8%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{a}} \]
   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-/l* 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 82.6

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

   7. Applied egg-rr 82.6%

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

   if 6.5e12 < t

   1. Initial program 71.0%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 82.6

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

   7. Simplified 82.6%

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

   8. Taylor expanded in t around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 77.7

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

   10. Simplified 77.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-118}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 6500000000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(a, \frac{y}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 86000000000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.45e-118)
   (+ x y)
   (if (<= t 86000000000000.0) (fma z (/ y a) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.45e-118) {
		tmp = x + y;
	} else if (t <= 86000000000000.0) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.45e-118)
		tmp = Float64(x + y);
	elseif (t <= 86000000000000.0)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.45e-118], N[(x + y), $MachinePrecision], If[LessEqual[t, 86000000000000.0], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.4499999999999999e-118 or 8.6e13 < t

   1. Initial program 74.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 73.4

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

   5. Simplified 73.4%

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

   if -1.4499999999999999e-118 < t < 8.6e13

   1. Initial program 91.9%

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

   3. Taylor expanded in t around 0

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

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

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

      2. *-lowering-*.f64 77.8

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

   5. Simplified 77.8%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{a}} \]
   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-/l* 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 82.6

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

   7. Applied egg-rr 82.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 86000000000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-118}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.6e-118) (+ x y) (if (<= t 2.5e+14) (fma y (/ z a) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.6e-118) {
		tmp = x + y;
	} else if (t <= 2.5e+14) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.6e-118)
		tmp = Float64(x + y);
	elseif (t <= 2.5e+14)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.6e-118], N[(x + y), $MachinePrecision], If[LessEqual[t, 2.5e+14], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.60000000000000002e-118 or 2.5e14 < t

   1. Initial program 74.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 73.4

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

   5. Simplified 73.4%

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

   if -1.60000000000000002e-118 < t < 2.5e14

   1. Initial program 91.9%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 82.5

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

   5. Simplified 82.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-118}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. --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. Add Preprocessing

Alternative 10: 96.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. --lowering--.f64 96.9

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

4. Applied egg-rr 96.9%

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

Alternative 11: 54.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+118}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+152}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -7.5e+118) y (if (<= y 2.6e+152) x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -7.5e+118) {
		tmp = y;
	} else if (y <= 2.6e+152) {
		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) :: tmp
    if (y <= (-7.5d+118)) then
        tmp = y
    else if (y <= 2.6d+152) 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 tmp;
	if (y <= -7.5e+118) {
		tmp = y;
	} else if (y <= 2.6e+152) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -7.5e+118:
		tmp = y
	elif y <= 2.6e+152:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -7.5e+118)
		tmp = y;
	elseif (y <= 2.6e+152)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -7.5e+118)
		tmp = y;
	elseif (y <= 2.6e+152)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -7.5e+118], y, If[LessEqual[y, 2.6e+152], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -7.50000000000000003e118 or 2.6000000000000001e152 < y

   1. Initial program 50.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 47.6

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

   5. Simplified 47.6%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 40.5%

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

   if -7.50000000000000003e118 < y < 2.6000000000000001e152

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 62.2%

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

Alternative 12: 61.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+207}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= a -1.15e+207) x (+ x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.15e+207) {
		tmp = x;
	} else {
		tmp = x + 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 (a <= (-1.15d+207)) then
        tmp = x
    else
        tmp = x + 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 (a <= -1.15e+207) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.15e+207:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.15e+207)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.15e+207)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.15e+207], x, N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.14999999999999997e207

   1. Initial program 69.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 82.6%

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

   if -1.14999999999999997e207 < a

   1. Initial program 81.9%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 64.4

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

   5. Simplified 64.4%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+207}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.1% 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 80.8%

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

3. Taylor expanded in x around inf

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

5. Simplified 46.9%

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

Developer Target 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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