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

Percentage Accurate: 85.8% → 98.3%

Time: 8.9s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. --lowering--.f64 97.5

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

4. Applied egg-rr 97.5%

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

Alternative 2: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-119}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.1e+64)
   (fma (/ (- t z) a) y x)
   (if (<= a 2.15e-119) (fma y (- 1.0 (/ t z)) x) (fma t (/ y a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.1e+64) {
		tmp = fma(((t - z) / a), y, x);
	} else if (a <= 2.15e-119) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.1e+64)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (a <= 2.15e-119)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.1e+64], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[a, 2.15e-119], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.10000000000000001e64

   1. Initial program 85.9%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

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

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. --lowering--.f64 95.9

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

   7. Simplified 95.9%

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

   if -1.10000000000000001e64 < a < 2.15e-119

   1. Initial program 90.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

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

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

      7. /-lowering-/.f64 87.3

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

   5. Simplified 87.3%

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

   if 2.15e-119 < a

   1. Initial program 87.9%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 81.0

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

   7. Simplified 81.0%

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

Alternative 3: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-119}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.6e+109)
   (fma y (/ t a) x)
   (if (<= a 2.15e-119) (fma y (- 1.0 (/ t z)) x) (fma t (/ y a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.6e+109) {
		tmp = fma(y, (t / a), x);
	} else if (a <= 2.15e-119) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.6e+109)
		tmp = fma(y, Float64(t / a), x);
	elseif (a <= 2.15e-119)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.6e+109], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 2.15e-119], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.6000000000000001e109

   1. Initial program 85.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 92.6

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

   5. Simplified 92.6%

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

   if -1.6000000000000001e109 < a < 2.15e-119

   1. Initial program 90.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

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

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

      7. /-lowering-/.f64 86.6

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

   5. Simplified 86.6%

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

   if 2.15e-119 < a

   1. Initial program 87.9%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 81.0

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

   7. Simplified 81.0%

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

Alternative 4: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.000125:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -0.000125) (+ y x) (if (<= z 1.65e-32) (fma y (/ t a) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.000125) {
		tmp = y + x;
	} else if (z <= 1.65e-32) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.000125)
		tmp = Float64(y + x);
	elseif (z <= 1.65e-32)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -0.000125], N[(y + x), $MachinePrecision], If[LessEqual[z, 1.65e-32], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e-4 or 1.65000000000000013e-32 < z

   1. Initial program 82.6%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 77.2

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

   5. Simplified 77.2%

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

   if -1.25e-4 < z < 1.65000000000000013e-32

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 77.2

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

   5. Simplified 77.2%

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

Alternative 5: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e-6) (+ y x) (if (<= z 1.65e-32) (fma t (/ y a) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e-6) {
		tmp = y + x;
	} else if (z <= 1.65e-32) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e-6)
		tmp = Float64(y + x);
	elseif (z <= 1.65e-32)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e-6], N[(y + x), $MachinePrecision], If[LessEqual[z, 1.65e-32], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1999999999999999e-6 or 1.65000000000000013e-32 < z

   1. Initial program 82.6%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 77.2

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

   5. Simplified 77.2%

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

   if -3.1999999999999999e-6 < z < 1.65000000000000013e-32

   1. Initial program 93.7%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 95.4

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 75.9

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

   7. Simplified 75.9%

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

Alternative 6: 96.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.1e+173) (fma y (- 1.0 (/ t z)) x) (fma (/ y (- z a)) (- z t) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.1e+173) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else {
		tmp = fma((y / (z - a)), (z - t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.1e+173)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	else
		tmp = fma(Float64(y / Float64(z - a)), Float64(z - t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.1e+173], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e173

   1. Initial program 80.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

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

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

      7. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if -1.1e173 < z

   1. Initial program 89.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 96.1

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

   4. Applied egg-rr 96.1%

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

Alternative 7: 63.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+124}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+101}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.5e+124) x (if (<= a 4e+101) (+ y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.5e+124) {
		tmp = x;
	} else if (a <= 4e+101) {
		tmp = y + x;
	} else {
		tmp = 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 (a <= (-3.5d+124)) then
        tmp = x
    else if (a <= 4d+101) then
        tmp = y + x
    else
        tmp = 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 (a <= -3.5e+124) {
		tmp = x;
	} else if (a <= 4e+101) {
		tmp = y + x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -3.5e+124:
		tmp = x
	elif a <= 4e+101:
		tmp = y + x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3.5e+124)
		tmp = x;
	elseif (a <= 4e+101)
		tmp = Float64(y + x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -3.5e+124)
		tmp = x;
	elseif (a <= 4e+101)
		tmp = y + x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3.5e+124], x, If[LessEqual[a, 4e+101], N[(y + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5000000000000001e124 or 3.9999999999999999e101 < a

   1. Initial program 84.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.6%

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

   if -3.5000000000000001e124 < a < 3.9999999999999999e101

   1. Initial program 90.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 67.4

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

   5. Simplified 67.4%

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

Alternative 8: 52.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= y 1.25e+105) x y))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 1.25e+105) {
		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 <= 1.25d+105) 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 <= 1.25e+105) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= 1.25e+105:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 1.25e+105)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= 1.25e+105)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, 1.25e+105], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 1.25000000000000011e105

   1. Initial program 90.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 62.2%

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

   if 1.25000000000000011e105 < y

   1. Initial program 71.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.7

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

   5. Simplified 42.7%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 39.5%

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

Alternative 9: 51.3% 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 88.6%

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

3. Taylor expanded in x around inf

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

5. Simplified 56.2%

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

Developer Target 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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