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

Percentage Accurate: 85.4% → 96.9%

Time: 10.5s

Alternatives: 12

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: 85.4% 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: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, \left(z - t\right) \cdot \frac{1}{t - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7e-142)
   (fma (/ y (- a t)) (- z t) x)
   (fma (- 0.0 y) (* (- z t) (/ 1.0 (- t a))) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7e-142) {
		tmp = fma((y / (a - t)), (z - t), x);
	} else {
		tmp = fma((0.0 - y), ((z - t) * (1.0 / (t - a))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7e-142)
		tmp = fma(Float64(y / Float64(a - t)), Float64(z - t), x);
	else
		tmp = fma(Float64(0.0 - y), Float64(Float64(z - t) * Float64(1.0 / Float64(t - a))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7e-142], N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.0 - y), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] * N[(1.0 / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.00000000000000029e-142

   1. Initial program 84.1%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 98.6

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

   4. Applied egg-rr 98.6%

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

   if 7.00000000000000029e-142 < t

   1. Initial program 85.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - y, \left(z - t\right) \cdot \frac{1}{t - a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -26500000000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+168}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -26500000000.0)
   (+ y x)
   (if (<= t 4.5e-96)
     (fma z (/ y a) x)
     (if (<= t 6.8e+168) (- x (/ (* y z) t)) (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -26500000000.0) {
		tmp = y + x;
	} else if (t <= 4.5e-96) {
		tmp = fma(z, (y / a), x);
	} else if (t <= 6.8e+168) {
		tmp = x - ((y * z) / t);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -26500000000.0)
		tmp = Float64(y + x);
	elseif (t <= 4.5e-96)
		tmp = fma(z, Float64(y / a), x);
	elseif (t <= 6.8e+168)
		tmp = Float64(x - Float64(Float64(y * z) / t));
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -26500000000.0], N[(y + x), $MachinePrecision], If[LessEqual[t, 4.5e-96], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 6.8e+168], N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.65e10 or 6.80000000000000005e168 < t

   1. Initial program 73.6%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 75.5

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

   5. Simplified 75.5%

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

   if -2.65e10 < t < 4.5e-96

   1. Initial program 92.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 93.8

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

   4. Applied egg-rr 93.8%

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

   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 82.7

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

   7. Simplified 82.7%

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

   if 4.5e-96 < t < 6.80000000000000005e168

   1. Initial program 89.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. *-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 94.4

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

   4. Applied egg-rr 94.4%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 77.9%

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

   8. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. *-lowering-*.f64 72.5

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

   10. Simplified 72.5%

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

Alternative 3: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a - t}, z, x\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y (- a t)) z x)))
   (if (<= z -2e+76) t_1 (if (<= z 3.9e-8) (fma y (/ t (- t a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / (a - t)), z, x);
	double tmp;
	if (z <= -2e+76) {
		tmp = t_1;
	} else if (z <= 3.9e-8) {
		tmp = fma(y, (t / (t - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / Float64(a - t)), z, x)
	tmp = 0.0
	if (z <= -2e+76)
		tmp = t_1;
	elseif (z <= 3.9e-8)
		tmp = fma(y, Float64(t / Float64(t - a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -2e+76], t$95$1, If[LessEqual[z, 3.9e-8], N[(y * N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000001e76 or 3.89999999999999985e-8 < z

   1. Initial program 85.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. *-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 96.8

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 92.6%

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

   if -2.0000000000000001e76 < z < 3.89999999999999985e-8

   1. Initial program 84.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. --lowering--.f64 90.3

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

   7. Simplified 90.3%

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

Alternative 4: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.2e-17)
   (fma y (/ t (- t a)) x)
   (if (<= t 4.8e-85) (fma y (/ (- z t) 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 <= -1.2e-17) {
		tmp = fma(y, (t / (t - a)), x);
	} else if (t <= 4.8e-85) {
		tmp = fma(y, ((z - t) / 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 <= -1.2e-17)
		tmp = fma(y, Float64(t / Float64(t - a)), x);
	elseif (t <= 4.8e-85)
		tmp = fma(y, Float64(Float64(z - t) / 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, -1.2e-17], N[(y * N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 4.8e-85], N[(y * N[(N[(z - t), $MachinePrecision] / 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 < -1.19999999999999993e-17

   1. Initial program 76.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. --lowering--.f64 82.1

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

   7. Simplified 82.1%

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

   if -1.19999999999999993e-17 < t < 4.8000000000000001e-85

   1. Initial program 91.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 87.2

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

   5. Simplified 87.2%

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

   if 4.8000000000000001e-85 < t

   1. Initial program 83.7%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 83.9

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

   5. Simplified 83.9%

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

Alternative 5: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7e-61)
   (fma y (/ t (- t a)) x)
   (if (<= t 1.6e-94) (fma z (/ 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 <= -7e-61) {
		tmp = fma(y, (t / (t - a)), x);
	} else if (t <= 1.6e-94) {
		tmp = fma(z, (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 <= -7e-61)
		tmp = fma(y, Float64(t / Float64(t - a)), x);
	elseif (t <= 1.6e-94)
		tmp = fma(z, 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, -7e-61], N[(y * N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.6e-94], N[(z * 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 < -7.0000000000000006e-61

   1. Initial program 78.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 97.7

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. --lowering--.f64 79.6

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

   7. Simplified 79.6%

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

   if -7.0000000000000006e-61 < t < 1.59999999999999998e-94

   1. Initial program 90.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 94.6

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

   4. Applied egg-rr 94.6%

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

   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 87.9

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

   7. Simplified 87.9%

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

   if 1.59999999999999998e-94 < t

   1. Initial program 84.1%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 83.1

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

   5. Simplified 83.1%

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

Alternative 6: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z t)) x)))
   (if (<= t -3.2e-32) t_1 (if (<= t 1.75e-94) (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 <= -3.2e-32) {
		tmp = t_1;
	} else if (t <= 1.75e-94) {
		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 <= -3.2e-32)
		tmp = t_1;
	elseif (t <= 1.75e-94)
		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, -3.2e-32], t$95$1, If[LessEqual[t, 1.75e-94], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2000000000000002e-32 or 1.74999999999999999e-94 < t

   1. Initial program 81.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 80.7

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

   5. Simplified 80.7%

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

   if -3.2000000000000002e-32 < t < 1.74999999999999999e-94

   1. Initial program 90.7%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 92.7

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

   4. Applied egg-rr 92.7%

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

   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 86.2

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

   7. Simplified 86.2%

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

Alternative 7: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -290000000000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -290000000000.0)
   (+ y x)
   (if (<= t 7.8e+56) (fma z (/ y a) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -290000000000.0) {
		tmp = y + x;
	} else if (t <= 7.8e+56) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -290000000000.0)
		tmp = Float64(y + x);
	elseif (t <= 7.8e+56)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -290000000000.0], N[(y + x), $MachinePrecision], If[LessEqual[t, 7.8e+56], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9e11 or 7.79999999999999989e56 < t

   1. Initial program 75.1%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 71.6

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

   5. Simplified 71.6%

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

   if -2.9e11 < t < 7.79999999999999989e56

   1. Initial program 92.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 95.1

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

   4. Applied egg-rr 95.1%

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

   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 76.1

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

   7. Simplified 76.1%

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

Alternative 8: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7800000000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7800000000.0)
   (+ y x)
   (if (<= t 5.9e+57) (fma y (/ z a) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7800000000.0) {
		tmp = y + x;
	} else if (t <= 5.9e+57) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7800000000.0)
		tmp = Float64(y + x);
	elseif (t <= 5.9e+57)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7800000000.0], N[(y + x), $MachinePrecision], If[LessEqual[t, 5.9e+57], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.8e9 or 5.90000000000000013e57 < t

   1. Initial program 75.1%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 71.6

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

   5. Simplified 71.6%

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

   if -7.8e9 < t < 5.90000000000000013e57

   1. Initial program 92.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 73.8

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

   5. Simplified 73.8%

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

Alternative 9: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \leq 2 \cdot 10^{+175}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (* y (- z t)) (- a t)) 2e+175) x y))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((y * (z - t)) / (a - t)) <= 2e+175) {
		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 * (z - t)) / (a - t)) <= 2d+175) 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 * (z - t)) / (a - t)) <= 2e+175) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((y * (z - t)) / (a - t)) <= 2e+175:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(y * Float64(z - t)) / Float64(a - t)) <= 2e+175)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((y * (z - t)) / (a - t)) <= 2e+175)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], 2e+175], x, y]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < 1.9999999999999999e175

   1. Initial program 93.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 58.2%

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

   if 1.9999999999999999e175 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))

   1. Initial program 37.3%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 38.9

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

   5. Simplified 38.9%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 33.0%

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

Alternative 10: 95.6% 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 84.5%

   \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
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 96.6

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

4. Applied egg-rr 96.6%

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

Alternative 11: 63.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.62 \cdot 10^{+31}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.62e+31) (+ y x) (if (<= t 3.9e+31) x (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.62e+31) {
		tmp = y + x;
	} else if (t <= 3.9e+31) {
		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 (t <= (-1.62d+31)) then
        tmp = y + x
    else if (t <= 3.9d+31) 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 (t <= -1.62e+31) {
		tmp = y + x;
	} else if (t <= 3.9e+31) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.62e+31:
		tmp = y + x
	elif t <= 3.9e+31:
		tmp = x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.62e+31)
		tmp = Float64(y + x);
	elseif (t <= 3.9e+31)
		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 (t <= -1.62e+31)
		tmp = y + x;
	elseif (t <= 3.9e+31)
		tmp = x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.62e+31], N[(y + x), $MachinePrecision], If[LessEqual[t, 3.9e+31], x, N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.6199999999999999e31 or 3.89999999999999999e31 < t

   1. Initial program 74.2%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 69.4

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

   5. Simplified 69.4%

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

   if -1.6199999999999999e31 < t < 3.89999999999999999e31

   1. Initial program 93.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 54.8%

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

Alternative 12: 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 84.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 51.2%

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

Developer Target 1: 98.3% 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 2024195 
(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))))