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

Percentage Accurate: 76.8% → 89.8%

Time: 8.1s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{t - z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{fma}\left(\frac{a - z}{t}, a, a - z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.15e+99)
   (fma (- 1.0 (/ (- t z) (- t a))) y x)
   (- x (/ y (/ t (fma (/ (- a z) t) a (- a z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.15e+99) {
		tmp = fma((1.0 - ((t - z) / (t - a))), y, x);
	} else {
		tmp = x - (y / (t / fma(((a - z) / t), a, (a - z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.15e+99)
		tmp = fma(Float64(1.0 - Float64(Float64(t - z) / Float64(t - a))), y, x);
	else
		tmp = Float64(x - Float64(y / Float64(t / fma(Float64(Float64(a - z) / t), a, Float64(a - z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.15e+99], N[(N[(1.0 - N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x - N[(y / N[(t / N[(N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] * a + N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1500000000000001e99

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if 2.1500000000000001e99 < t

   1. Initial program 59.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{a \cdot y}{t} + -1 \cdot \frac{a \cdot \left(-1 \cdot \left(y \cdot z\right) - -1 \cdot \left(a \cdot y\right)\right)}{{t}^{2}}\right)\right) - -1 \cdot \frac{y \cdot z}{t}} \]

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.3%

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

   10. Applied rewrites 90.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{t - z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{fma}\left(\frac{a - z}{t}, a, a - z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{t - z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(a, \frac{a - z}{t}, a\right) - z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.15e+99)
   (fma (- 1.0 (/ (- t z) (- t a))) y x)
   (- x (* (/ (- (fma a (/ (- a z) t) a) z) t) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.15e+99) {
		tmp = fma((1.0 - ((t - z) / (t - a))), y, x);
	} else {
		tmp = x - (((fma(a, ((a - z) / t), a) - z) / t) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.15e+99)
		tmp = fma(Float64(1.0 - Float64(Float64(t - z) / Float64(t - a))), y, x);
	else
		tmp = Float64(x - Float64(Float64(Float64(fma(a, Float64(Float64(a - z) / t), a) - z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.15e+99], N[(N[(1.0 - N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x - N[(N[(N[(N[(a * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] + a), $MachinePrecision] - z), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1500000000000001e99

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if 2.1500000000000001e99 < t

   1. Initial program 59.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{a \cdot y}{t} + -1 \cdot \frac{a \cdot \left(-1 \cdot \left(y \cdot z\right) - -1 \cdot \left(a \cdot y\right)\right)}{{t}^{2}}\right)\right) - -1 \cdot \frac{y \cdot z}{t}} \]

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{t - z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(a, \frac{a - z}{t}, a\right) - z}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{t - z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{a + t}{t} \cdot \frac{a - z}{t}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.15e+99)
   (fma (- 1.0 (/ (- t z) (- t a))) y x)
   (- x (* (* (/ (+ a t) t) (/ (- a z) t)) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.15e+99) {
		tmp = fma((1.0 - ((t - z) / (t - a))), y, x);
	} else {
		tmp = x - ((((a + t) / t) * ((a - z) / t)) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.15e+99)
		tmp = fma(Float64(1.0 - Float64(Float64(t - z) / Float64(t - a))), y, x);
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a + t) / t) * Float64(Float64(a - z) / t)) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.15e+99], N[(N[(1.0 - N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x - N[(N[(N[(N[(a + t), $MachinePrecision] / t), $MachinePrecision] * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1500000000000001e99

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if 2.1500000000000001e99 < t

   1. Initial program 59.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{a \cdot y}{t} + -1 \cdot \frac{a \cdot \left(-1 \cdot \left(y \cdot z\right) - -1 \cdot \left(a \cdot y\right)\right)}{{t}^{2}}\right)\right) - -1 \cdot \frac{y \cdot z}{t}} \]

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in t around 0

      \[\leadsto x - y \cdot \frac{a \cdot \left(a - z\right) + t \cdot \left(a - z\right)}{{t}^{\color{blue}{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{t - z}{t - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{a + t}{t} \cdot \frac{a - z}{t}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{-27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z a) t) x)))
   (if (<= t -6.8e-27) t_1 (if (<= t 2.1e-24) (fma (- 1.0 (/ z a)) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((z - a) / t), x);
	double tmp;
	if (t <= -6.8e-27) {
		tmp = t_1;
	} else if (t <= 2.1e-24) {
		tmp = fma((1.0 - (z / a)), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(z - a) / t), x)
	tmp = 0.0
	if (t <= -6.8e-27)
		tmp = t_1;
	elseif (t <= 2.1e-24)
		tmp = fma(Float64(1.0 - 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[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -6.8e-27], t$95$1, If[LessEqual[t, 2.1e-24], N[(N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.7999999999999994e-27 or 2.0999999999999999e-24 < t

   1. Initial program 64.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.7%

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

   9. Taylor expanded in t around -inf

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

   11. Applied rewrites 82.3%

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

   if -6.7999999999999994e-27 < t < 2.0999999999999999e-24

   1. Initial program 91.4%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 90.9%

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

Alternative 5: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{+41}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.9e+41)
   (+ y x)
   (if (<= a 6.2e+80) (fma y (/ (- z a) t) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.9e+41) {
		tmp = y + x;
	} else if (a <= 6.2e+80) {
		tmp = fma(y, ((z - a) / t), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3.9e+41)
		tmp = Float64(y + x);
	elseif (a <= 6.2e+80)
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3.9e+41], N[(y + x), $MachinePrecision], If[LessEqual[a, 6.2e+80], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8999999999999997e41 or 6.19999999999999976e80 < a

   1. Initial program 77.5%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. lower-+.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if -3.8999999999999997e41 < a < 6.19999999999999976e80

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in t around -inf

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

   11. Applied rewrites 74.3%

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

Alternative 6: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{+40}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.15e+40) (+ y x) (if (<= a 6.2e+80) (fma (/ z t) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.15e+40) {
		tmp = y + x;
	} else if (a <= 6.2e+80) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.15e+40)
		tmp = Float64(y + x);
	elseif (a <= 6.2e+80)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.15e+40], N[(y + x), $MachinePrecision], If[LessEqual[a, 6.2e+80], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.1500000000000001e40 or 6.19999999999999976e80 < a

   1. Initial program 77.5%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. lower-+.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if -2.1500000000000001e40 < a < 6.19999999999999976e80

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.2%

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

Alternative 7: 89.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 - \frac{t - z}{t - a}, y, x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma (- 1.0 (/ (- t z) (- t a))) y x))

C

double code(double x, double y, double z, double t, double a) {
	return fma((1.0 - ((t - z) / (t - a))), y, x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(1.0 - Float64(Float64(t - z) / Float64(t - a))), y, x)
end

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. sub-neg N/A

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

   4. *-rgt-identity N/A

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

   5. associate-/l* N/A

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

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

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

   7. mul-1-neg N/A

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

   8. distribute-lft-in N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower--.f64 92.9

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

5. Applied rewrites 92.9%

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

6. Final simplification 92.9%

   \[\leadsto \mathsf{fma}\left(1 - \frac{t - z}{t - a}, y, x\right) \]
7. Add Preprocessing

Alternative 8: 63.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{-11}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.12e-11) (+ y x) (if (<= a 3.8e-63) (* 1.0 x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.12e-11) {
		tmp = y + x;
	} else if (a <= 3.8e-63) {
		tmp = 1.0 * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-1.12d-11)) then
        tmp = y + x
    else if (a <= 3.8d-63) then
        tmp = 1.0d0 * x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.12e-11) {
		tmp = y + x;
	} else if (a <= 3.8e-63) {
		tmp = 1.0 * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.12e-11:
		tmp = y + x
	elif a <= 3.8e-63:
		tmp = 1.0 * x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.12e-11)
		tmp = Float64(y + x);
	elseif (a <= 3.8e-63)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.12e-11)
		tmp = y + x;
	elseif (a <= 3.8e-63)
		tmp = 1.0 * x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.12e-11], N[(y + x), $MachinePrecision], If[LessEqual[a, 3.8e-63], N[(1.0 * x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1200000000000001e-11 or 3.80000000000000017e-63 < a

   1. Initial program 79.2%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 87.0%

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

   9. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. lower-+.f64 74.4

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

   11. Applied rewrites 74.4%

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

   if -1.1200000000000001e-11 < a < 3.80000000000000017e-63

   1. Initial program 75.2%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.3%

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

   9. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 60.4% accurate, 7.3× 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 77.4%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. sub-neg N/A

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

   4. *-rgt-identity N/A

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

   5. associate-/l* N/A

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

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

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

   7. mul-1-neg N/A

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

   8. distribute-lft-in N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower--.f64 92.9

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

5. Applied rewrites 92.9%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 70.7%

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

9. Taylor expanded in a around inf

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

    1. +-commutative N/A

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

    2. lower-+.f64 63.0

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

11. Applied rewrites 63.0%

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

Developer Target 1: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 < -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (< t_2 -1.3664970889390727e-7)
     t_1
     (if (< t_2 1.4754293444577233e-239)
       (/ (- (* y (- a z)) (* x t)) (- a t))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 < -1.3664970889390727e-7) {
		tmp = t_1;
	} else if (t_2 < 1.4754293444577233e-239) {
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y + x) - (((z - t) * (1.0d0 / (a - t))) * y)
    t_2 = (x + y) - (((z - t) * y) / (a - t))
    if (t_2 < (-1.3664970889390727d-7)) then
        tmp = t_1
    else if (t_2 < 1.4754293444577233d-239) then
        tmp = ((y * (a - z)) - (x * t)) / (a - t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 < -1.3664970889390727e-7) {
		tmp = t_1;
	} else if (t_2 < 1.4754293444577233e-239) {
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y)
	t_2 = (x + y) - (((z - t) * y) / (a - t))
	tmp = 0
	if t_2 < -1.3664970889390727e-7:
		tmp = t_1
	elif t_2 < 1.4754293444577233e-239:
		tmp = ((y * (a - z)) - (x * t)) / (a - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y + x) - Float64(Float64(Float64(z - t) * Float64(1.0 / Float64(a - t))) * y))
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 < -1.3664970889390727e-7)
		tmp = t_1;
	elseif (t_2 < 1.4754293444577233e-239)
		tmp = Float64(Float64(Float64(y * Float64(a - z)) - Float64(x * t)) / Float64(a - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	t_2 = (x + y) - (((z - t) * y) / (a - t));
	tmp = 0.0;
	if (t_2 < -1.3664970889390727e-7)
		tmp = t_1;
	elseif (t_2 < 1.4754293444577233e-239)
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -1.3664970889390727e-7], t$95$1, If[Less[t$95$2, 1.4754293444577233e-239], N[(N[(N[(y * N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(x * t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -13664970889390727/100000000000000000000000) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 14754293444577233/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)))))

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