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

Percentage Accurate: 85.4% → 99.2%

Time: 9.8s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.4% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ t_2 := \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+297}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+201}:\\ \;\;\;\;x + t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y (- a t)) (- z t) x)) (t_2 (/ (* y (- z t)) (- a t))))
   (if (<= t_2 -1e+297) t_1 (if (<= t_2 5e+201) (+ x t_2) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / (a - t)), (z - t), x);
	double t_2 = (y * (z - t)) / (a - t);
	double tmp;
	if (t_2 <= -1e+297) {
		tmp = t_1;
	} else if (t_2 <= 5e+201) {
		tmp = x + t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / Float64(a - t)), Float64(z - t), x)
	t_2 = Float64(Float64(y * Float64(z - t)) / Float64(a - t))
	tmp = 0.0
	if (t_2 <= -1e+297)
		tmp = t_1;
	elseif (t_2 <= 5e+201)
		tmp = Float64(x + t_2);
	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] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+297], t$95$1, If[LessEqual[t$95$2, 5e+201], N[(x + t$95$2), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < -1e297 or 4.9999999999999995e201 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))

   1. Initial program 51.3%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   if -1e297 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < 4.9999999999999995e201

   1. Initial program 99.9%

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

Alternative 2: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, -t, x\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.5e-38)
   (fma (/ y (- a t)) (- t) x)
   (if (<= t 1.25e-79) (fma (/ y a) (- z t) 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 <= -3.5e-38) {
		tmp = fma((y / (a - t)), -t, x);
	} else if (t <= 1.25e-79) {
		tmp = fma((y / a), (z - t), 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 <= -3.5e-38)
		tmp = fma(Float64(y / Float64(a - t)), Float64(-t), x);
	elseif (t <= 1.25e-79)
		tmp = fma(Float64(y / a), Float64(z - t), 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, -3.5e-38], N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * (-t) + x), $MachinePrecision], If[LessEqual[t, 1.25e-79], N[(N[(y / a), $MachinePrecision] * N[(z - t), $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 < -3.5000000000000001e-38

   1. Initial program 76.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-lowering-neg.f64 88.7

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

   5. Simplified 88.7%

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

   if -3.5000000000000001e-38 < t < 1.25e-79

   1. Initial program 96.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 84.4

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

   5. Simplified 84.4%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. flip3-- N/A

         \[\leadsto \frac{y}{a} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{z}^{3} - {t}^{3}}{z \cdot z + \left(t \cdot t + z \cdot t\right)}}}} + x \]

      6. clear-num N/A

         \[\leadsto \frac{y}{a} \cdot \frac{1}{\color{blue}{\frac{z \cdot z + \left(t \cdot t + z \cdot t\right)}{{z}^{3} - {t}^{3}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y}{a} \cdot \color{blue}{\frac{{z}^{3} - {t}^{3}}{z \cdot z + \left(t \cdot t + z \cdot t\right)}} + x \]

      8. flip3-- N/A

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

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

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

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

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

      11. --lowering--.f64 86.4

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

   7. Applied egg-rr 86.4%

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

   if 1.25e-79 < t

   1. Initial program 81.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 84.1

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

   5. Simplified 84.1%

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

Alternative 3: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{if}\;a \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z t) a) x)))
   (if (<= a -4.4e-5) t_1 (if (<= a 4.1e-10) (fma y (- 1.0 (/ z t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((z - t) / a), x);
	double tmp;
	if (a <= -4.4e-5) {
		tmp = t_1;
	} else if (a <= 4.1e-10) {
		tmp = fma(y, (1.0 - (z / t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(z - t) / a), x)
	tmp = 0.0
	if (a <= -4.4e-5)
		tmp = t_1;
	elseif (a <= 4.1e-10)
		tmp = fma(y, Float64(1.0 - Float64(z / t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -4.4e-5], t$95$1, If[LessEqual[a, 4.1e-10], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.3999999999999999e-5 or 4.0999999999999998e-10 < a

   1. Initial program 87.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 89.0

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

   5. Simplified 89.0%

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

   if -4.3999999999999999e-5 < a < 4.0999999999999998e-10

   1. Initial program 86.2%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 82.1

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

   5. Simplified 82.1%

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

Alternative 4: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{if}\;t \leq -7 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z t)) x)))
   (if (<= t -7e-130) t_1 (if (<= t 4.5e-80) (fma y (/ z a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (z / t)), x);
	double tmp;
	if (t <= -7e-130) {
		tmp = t_1;
	} else if (t <= 4.5e-80) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(z / t)), x)
	tmp = 0.0
	if (t <= -7e-130)
		tmp = t_1;
	elseif (t <= 4.5e-80)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -7e-130], t$95$1, If[LessEqual[t, 4.5e-80], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.9999999999999998e-130 or 4.5000000000000003e-80 < t

   1. Initial program 81.9%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 77.4

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

   5. Simplified 77.4%

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

   if -6.9999999999999998e-130 < t < 4.5000000000000003e-80

   1. Initial program 95.6%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 85.8

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

   5. Simplified 85.8%

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

Alternative 5: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

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

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

   7. --lowering--.f64 97.8

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

4. Applied egg-rr 97.8%

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

Alternative 6: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+97}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 7.8:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.2e+97) (+ x y) (if (<= t 7.8) (fma y (/ z a) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.2e+97) {
		tmp = x + y;
	} else if (t <= 7.8) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.2e+97)
		tmp = Float64(x + y);
	elseif (t <= 7.8)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.2e+97], N[(x + y), $MachinePrecision], If[LessEqual[t, 7.8], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.20000000000000016e97 or 7.79999999999999982 < t

   1. Initial program 71.4%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 80.6

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

   5. Simplified 80.6%

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

   if -3.20000000000000016e97 < t < 7.79999999999999982

   1. Initial program 96.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 74.3

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

   5. Simplified 74.3%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+97}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 7.8:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-137}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -6.6e-137)
   (+ x y)
   (if (<= x 2.8e-208) (/ (* y z) a) (fma t (/ y t) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -6.6e-137) {
		tmp = x + y;
	} else if (x <= 2.8e-208) {
		tmp = (y * z) / a;
	} else {
		tmp = fma(t, (y / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -6.6e-137)
		tmp = Float64(x + y);
	elseif (x <= 2.8e-208)
		tmp = Float64(Float64(y * z) / a);
	else
		tmp = fma(t, Float64(y / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -6.6e-137], N[(x + y), $MachinePrecision], If[LessEqual[x, 2.8e-208], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], N[(t * N[(y / t), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6000000000000004e-137

   1. Initial program 85.8%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 64.3

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

   5. Simplified 64.3%

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

   if -6.6000000000000004e-137 < x < 2.80000000000000001e-208

   1. Initial program 94.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 49.9

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

   5. Simplified 49.9%

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

   6. Taylor expanded in y around inf

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

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

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

      2. *-lowering-*.f64 48.3

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

   8. Simplified 48.3%

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

   if 2.80000000000000001e-208 < x

   1. Initial program 82.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot \left(z - t\right)}{t}} \]
   6. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. remove-double-neg N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

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

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

      15. /-lowering-/.f64 72.3

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

   7. Simplified 72.3%

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

   8. Taylor expanded in t around inf

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

   10. Simplified 75.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-137}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-138}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-212}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -4.7e-138) (+ x y) (if (<= x 4.1e-212) (/ (* y z) a) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -4.7e-138) {
		tmp = x + y;
	} else if (x <= 4.1e-212) {
		tmp = (y * z) / a;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-4.7d-138)) then
        tmp = x + y
    else if (x <= 4.1d-212) then
        tmp = (y * z) / a
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -4.7e-138) {
		tmp = x + y;
	} else if (x <= 4.1e-212) {
		tmp = (y * z) / a;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -4.7e-138:
		tmp = x + y
	elif x <= 4.1e-212:
		tmp = (y * z) / a
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -4.7e-138)
		tmp = Float64(x + y);
	elseif (x <= 4.1e-212)
		tmp = Float64(Float64(y * z) / a);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -4.7e-138)
		tmp = x + y;
	elseif (x <= 4.1e-212)
		tmp = (y * z) / a;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -4.7e-138], N[(x + y), $MachinePrecision], If[LessEqual[x, 4.1e-212], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000001e-138 or 4.10000000000000014e-212 < x

   1. Initial program 84.2%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 68.0

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

   5. Simplified 68.0%

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

   if -4.7000000000000001e-138 < x < 4.10000000000000014e-212

   1. Initial program 94.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 49.9

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

   5. Simplified 49.9%

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

   6. Taylor expanded in y around inf

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

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

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

      2. *-lowering-*.f64 48.3

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

   8. Simplified 48.3%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-138}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-212}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-136}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-208}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.55e-136) (+ x y) (if (<= x 1.7e-208) (* z (/ y a)) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.55e-136) {
		tmp = x + y;
	} else if (x <= 1.7e-208) {
		tmp = z * (y / a);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-1.55d-136)) then
        tmp = x + y
    else if (x <= 1.7d-208) then
        tmp = z * (y / a)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.55e-136) {
		tmp = x + y;
	} else if (x <= 1.7e-208) {
		tmp = z * (y / a);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.55e-136:
		tmp = x + y
	elif x <= 1.7e-208:
		tmp = z * (y / a)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.55e-136)
		tmp = Float64(x + y);
	elseif (x <= 1.7e-208)
		tmp = Float64(z * Float64(y / a));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.55e-136)
		tmp = x + y;
	elseif (x <= 1.7e-208)
		tmp = z * (y / a);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.55e-136], N[(x + y), $MachinePrecision], If[LessEqual[x, 1.7e-208], N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55e-136 or 1.7e-208 < x

   1. Initial program 84.2%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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 68.0

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

   5. Simplified 68.0%

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

   if -1.55e-136 < x < 1.7e-208

   1. Initial program 94.8%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. --lowering--.f64 65.2

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

   5. Simplified 65.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 64.0

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

   7. Applied egg-rr 64.0%

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

   8. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 47.1

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

   10. Simplified 47.1%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-136}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-208}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.8% 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 86.6%

   \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
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 96.2

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

4. Applied egg-rr 96.2%

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

Alternative 11: 53.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-221}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-173}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -5e-221) x (if (<= x 1.75e-173) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -5e-221) {
		tmp = x;
	} else if (x <= 1.75e-173) {
		tmp = y;
	} 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 (x <= (-5d-221)) then
        tmp = x
    else if (x <= 1.75d-173) then
        tmp = y
    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 (x <= -5e-221) {
		tmp = x;
	} else if (x <= 1.75e-173) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -5e-221:
		tmp = x
	elif x <= 1.75e-173:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -5e-221)
		tmp = x;
	elseif (x <= 1.75e-173)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -5e-221)
		tmp = x;
	elseif (x <= 1.75e-173)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -5e-221], x, If[LessEqual[x, 1.75e-173], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999996e-221 or 1.75000000000000007e-173 < x

   1. Initial program 85.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 56.2%

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

   if -4.99999999999999996e-221 < x < 1.75000000000000007e-173

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. --lowering--.f64 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 28.6%

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

Alternative 12: 60.3% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 86.6%

   \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
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 57.5

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

5. Simplified 57.5%

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

6. Final simplification 57.5%

   \[\leadsto x + y \]
7. Add Preprocessing

Alternative 13: 50.3% 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 86.6%

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

3. Taylor expanded in x around inf

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

5. Simplified 46.2%

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

Developer Target 1: 98.4% accurate, 0.8× speedup

Math

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

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 2024199 
(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))))