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

Percentage Accurate: 85.7% → 98.5%

Time: 8.3s

Alternatives: 13

Speedup: 0.8×

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: 85.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - t} \cdot \left(z - t\right)\\ t_2 := \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 3 \cdot 10^{+264}:\\ \;\;\;\;t\_2 + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- a t)) (- z t))) (t_2 (/ (* (- z t) y) (- a t))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 3e+264) (+ t_2 x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (a - t)) * (z - t);
	double t_2 = ((z - t) * y) / (a - t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 3e+264) {
		tmp = t_2 + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (a - t)) * (z - t);
	double t_2 = ((z - t) * y) / (a - t);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 3e+264) {
		tmp = t_2 + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / (a - t)) * (z - t)
	t_2 = ((z - t) * y) / (a - t)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 3e+264:
		tmp = t_2 + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(a - t)) * Float64(z - t))
	t_2 = Float64(Float64(Float64(z - t) * y) / Float64(a - t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 3e+264)
		tmp = Float64(t_2 + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / (a - t)) * (z - t);
	t_2 = ((z - t) * y) / (a - t);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 3e+264)
		tmp = t_2 + x;
	else
		tmp = t_1;
	end
	tmp_2 = 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]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 3e+264], N[(t$95$2 + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < -inf.0 or 3e264 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))

   1. Initial program 38.0%

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

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

      2. div-sub N/A

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

      3. distribute-lft-out-- N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-out-- N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 91.1

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

   5. Applied rewrites 91.1%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < 3e264

   1. Initial program 99.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a - t} \leq -\infty:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a - t} \leq 3 \cdot 10^{+264}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - t} \cdot \left(z - t\right)\\ t_2 := \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- a t)) (- z t))) (t_2 (/ (* (- z t) y) (- a t))))
   (if (<= t_2 -5e+40) t_1 (if (<= t_2 4e+66) (fma y (/ t (- t a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (a - t)) * (z - t);
	double t_2 = ((z - t) * y) / (a - t);
	double tmp;
	if (t_2 <= -5e+40) {
		tmp = t_1;
	} else if (t_2 <= 4e+66) {
		tmp = fma(y, (t / (t - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(a - t)) * Float64(z - t))
	t_2 = Float64(Float64(Float64(z - t) * y) / Float64(a - t))
	tmp = 0.0
	if (t_2 <= -5e+40)
		tmp = t_1;
	elseif (t_2 <= 4e+66)
		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] * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+40], t$95$1, If[LessEqual[t$95$2, 4e+66], N[(y * N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < -5.00000000000000003e40 or 3.99999999999999978e66 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))

   1. Initial program 65.0%

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

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

      2. div-sub N/A

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

      3. distribute-lft-out-- N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-out-- N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -5.00000000000000003e40 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t)) < 3.99999999999999978e66

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{t \cdot y}{a - t}} \]
   6. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 88.3

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

   7. Applied rewrites 88.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a - t} \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a - t} \leq 4 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{a - t} + x\\ \mathbf{if}\;z \leq -1 \cdot 10^{+161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \mathbf{elif}\;z \leq 52000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) (- a t)) x)))
   (if (<= z -1e+161)
     t_1
     (if (<= z -1.15e+17)
       (* (/ y (- a t)) (- z t))
       (if (<= z 52000000000.0) (fma y (/ t (- t a)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z * y) / (a - t)) + x;
	double tmp;
	if (z <= -1e+161) {
		tmp = t_1;
	} else if (z <= -1.15e+17) {
		tmp = (y / (a - t)) * (z - t);
	} else if (z <= 52000000000.0) {
		tmp = fma(y, (t / (t - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z * y) / Float64(a - t)) + x)
	tmp = 0.0
	if (z <= -1e+161)
		tmp = t_1;
	elseif (z <= -1.15e+17)
		tmp = Float64(Float64(y / Float64(a - t)) * Float64(z - t));
	elseif (z <= 52000000000.0)
		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[(N[(z * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1e+161], t$95$1, If[LessEqual[z, -1.15e+17], N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 52000000000.0], N[(y * N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e161 or 5.2e10 < z

   1. Initial program 87.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if -1e161 < z < -1.15e17

   1. Initial program 70.5%

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

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

      2. div-sub N/A

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

      3. distribute-lft-out-- N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-out-- N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -1.15e17 < z < 5.2e10

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{t \cdot y}{a - t}} \]
   6. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 95.5

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

   7. Applied rewrites 95.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\frac{z \cdot y}{a - t} + x\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right)\\ \mathbf{elif}\;z \leq 52000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{a - t} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-16}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-71}:\\ \;\;\;\;\left(1 - \frac{z}{t}\right) \cdot y\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.9e-16)
   (+ y x)
   (if (<= t -1.75e-71)
     (* (- 1.0 (/ z t)) y)
     (if (<= t 5.8e+26) (fma (/ z a) y x) (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.9e-16) {
		tmp = y + x;
	} else if (t <= -1.75e-71) {
		tmp = (1.0 - (z / t)) * y;
	} else if (t <= 5.8e+26) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.9e-16)
		tmp = Float64(y + x);
	elseif (t <= -1.75e-71)
		tmp = Float64(Float64(1.0 - Float64(z / t)) * y);
	elseif (t <= 5.8e+26)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.9e-16], N[(y + x), $MachinePrecision], If[LessEqual[t, -1.75e-71], N[(N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 5.8e+26], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.90000000000000006e-16 or 5.8e26 < t

   1. Initial program 71.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. lower-+.f64 79.9

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

   5. Applied rewrites 79.9%

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

   if -1.90000000000000006e-16 < t < -1.75e-71

   1. Initial program 99.7%

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

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

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

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. associate-+l- N/A

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

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.4%

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

   if -1.75e-71 < t < 5.8e26

   1. Initial program 93.8%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 6: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.2e+16)
   (fma y (/ t (- t a)) x)
   (if (<= a 5e-15) (fma (- 1.0 (/ z t)) y x) (fma (- z t) (/ y a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.2e+16) {
		tmp = fma(y, (t / (t - a)), x);
	} else if (a <= 5e-15) {
		tmp = fma((1.0 - (z / t)), y, x);
	} else {
		tmp = fma((z - t), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.2e+16)
		tmp = fma(y, Float64(t / Float64(t - a)), x);
	elseif (a <= 5e-15)
		tmp = fma(Float64(1.0 - Float64(z / t)), y, x);
	else
		tmp = fma(Float64(z - t), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.2e+16], N[(y * N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 5e-15], N[(N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.2e16

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{t \cdot y}{a - t}} \]
   6. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if -4.2e16 < a < 4.99999999999999999e-15

   1. Initial program 84.8%

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

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

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

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. associate-+l- N/A

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

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 4.99999999999999999e-15 < a

   1. Initial program 78.1%

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 81.6

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

   5. Applied rewrites 81.6%

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

Alternative 7: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.2e+16)
   (fma y (/ t (- t a)) x)
   (if (<= a 3.3e-19) (fma (- 1.0 (/ z t)) y x) (fma (/ z a) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.2e+16) {
		tmp = fma(y, (t / (t - a)), x);
	} else if (a <= 3.3e-19) {
		tmp = fma((1.0 - (z / t)), y, x);
	} else {
		tmp = fma((z / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.2e+16)
		tmp = fma(y, Float64(t / Float64(t - a)), x);
	elseif (a <= 3.3e-19)
		tmp = fma(Float64(1.0 - Float64(z / t)), y, x);
	else
		tmp = fma(Float64(z / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.2e+16], N[(y * N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 3.3e-19], N[(N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.2e16

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{t \cdot y}{a - t}} \]
   6. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if -4.2e16 < a < 3.2999999999999998e-19

   1. Initial program 84.8%

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

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

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

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. associate-+l- N/A

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

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 3.2999999999999998e-19 < a

   1. Initial program 78.1%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.2

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

   5. Applied rewrites 79.2%

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

Alternative 8: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t}{t - a}, x\right)\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 18000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ t (- t a)) x)))
   (if (<= t -3.2e-115) t_1 (if (<= t 18000000000.0) (fma (/ z a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (t / (t - a)), x);
	double tmp;
	if (t <= -3.2e-115) {
		tmp = t_1;
	} else if (t <= 18000000000.0) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(t / Float64(t - a)), x)
	tmp = 0.0
	if (t <= -3.2e-115)
		tmp = t_1;
	elseif (t <= 18000000000.0)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -3.2e-115], t$95$1, If[LessEqual[t, 18000000000.0], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2e-115 or 1.8e10 < t

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{t \cdot y}{a - t}} \]
   6. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if -3.2e-115 < t < 1.8e10

   1. Initial program 93.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 75.6

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

   5. Applied rewrites 75.6%

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

Alternative 9: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-48}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.5e-48) (+ y x) (if (<= t 5.8e+26) (fma (/ z a) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.5e-48) {
		tmp = y + x;
	} else if (t <= 5.8e+26) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.5e-48)
		tmp = Float64(y + x);
	elseif (t <= 5.8e+26)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.5e-48], N[(y + x), $MachinePrecision], If[LessEqual[t, 5.8e+26], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.50000000000000042e-48 or 5.8e26 < t

   1. Initial program 72.6%

      \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
   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. lower-+.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -7.50000000000000042e-48 < t < 5.8e26

   1. Initial program 94.1%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

Alternative 10: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-114}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.35e-114) (+ y x) (if (<= t 8.5e+25) (fma z (/ y a) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.35e-114) {
		tmp = y + x;
	} else if (t <= 8.5e+25) {
		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 <= -1.35e-114)
		tmp = Float64(y + x);
	elseif (t <= 8.5e+25)
		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, -1.35e-114], N[(y + x), $MachinePrecision], If[LessEqual[t, 8.5e+25], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35e-114 or 8.5000000000000007e25 < t

   1. Initial program 75.5%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -1.35e-114 < t < 8.5000000000000007e25

   1. Initial program 93.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   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-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 72.8

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

   7. Applied rewrites 72.8%

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

Alternative 11: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+252}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.55e+252) (* (/ z a) y) (+ y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e+252) {
		tmp = (z / a) * y;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.55d+252)) then
        tmp = (z / a) * y
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e+252) {
		tmp = (z / a) * y;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.55e+252:
		tmp = (z / a) * y
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.55e+252)
		tmp = Float64(Float64(z / a) * y);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.55e+252)
		tmp = (z / a) * y;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.55e+252], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.54999999999999991e252

   1. Initial program 58.4%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 58.4%

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

   if -1.54999999999999991e252 < z

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 62.9

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

   5. Applied rewrites 62.9%

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

Alternative 12: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+252}:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.05e+252) (* (/ y a) z) (+ y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e+252) {
		tmp = (y / a) * z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.05d+252)) then
        tmp = (y / a) * z
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e+252) {
		tmp = (y / a) * z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.05e+252:
		tmp = (y / a) * z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.05e+252)
		tmp = Float64(Float64(y / a) * z);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.05e+252)
		tmp = (y / a) * z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.05e+252], N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0500000000000001e252

   1. Initial program 58.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

   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-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 68.1

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 37.7%

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

   12. Applied rewrites 58.2%

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

   if -1.0500000000000001e252 < z

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 62.9

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

   5. Applied rewrites 62.9%

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

Alternative 13: 61.0% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

   \[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
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. lower-+.f64 61.0

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

5. Applied rewrites 61.0%

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

Developer Target 1: 98.5% 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 2024296 
(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))))