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

Percentage Accurate: 76.7% → 90.7%

Time: 11.1s

Alternatives: 9

Speedup: 1.0×

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: 76.7% 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: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+77}:\\ \;\;\;\;x + \left(\frac{y}{t} \cdot \left(z - a\right)\right) \cdot 1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.5e+77)
   (+ x (* (* (/ y t) (- z a)) 1.0))
   (if (<= t 2.8e+122)
     (fma y (+ 1.0 (/ (- z t) (- t a))) x)
     (fma y (/ (- z a) t) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.5e+77) {
		tmp = x + (((y / t) * (z - a)) * 1.0);
	} else if (t <= 2.8e+122) {
		tmp = fma(y, (1.0 + ((z - t) / (t - a))), x);
	} else {
		tmp = fma(y, ((z - a) / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.5e+77)
		tmp = Float64(x + Float64(Float64(Float64(y / t) * Float64(z - a)) * 1.0));
	elseif (t <= 2.8e+122)
		tmp = fma(y, Float64(1.0 + Float64(Float64(z - t) / Float64(t - a))), x);
	else
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.5e+77], N[(x + N[(N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+122], N[(y * N[(1.0 + N[(N[(z - t), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.50000000000000024e77

   1. Initial program 64.0%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 93.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.7%

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

   if -4.50000000000000024e77 < t < 2.8e122

   1. Initial program 85.9%

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

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if 2.8e122 < t

   1. Initial program 46.4%

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

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

   8. Applied rewrites 96.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+77}:\\ \;\;\;\;x + \left(\frac{y}{t} \cdot \left(z - a\right)\right) \cdot 1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.5e+77)
   (fma (/ y t) (- z a) x)
   (if (<= t 2.8e+122)
     (fma y (+ 1.0 (/ (- z t) (- t a))) x)
     (fma y (/ (- z a) t) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.5e+77) {
		tmp = fma((y / t), (z - a), x);
	} else if (t <= 2.8e+122) {
		tmp = fma(y, (1.0 + ((z - t) / (t - a))), x);
	} else {
		tmp = fma(y, ((z - a) / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.5e+77)
		tmp = fma(Float64(y / t), Float64(z - a), x);
	elseif (t <= 2.8e+122)
		tmp = fma(y, Float64(1.0 + Float64(Float64(z - t) / Float64(t - a))), x);
	else
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.5e+77], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 2.8e+122], N[(y * N[(1.0 + N[(N[(z - t), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.50000000000000024e77

   1. Initial program 55.1%

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

   3. Taylor expanded in t around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if -4.50000000000000024e77 < t < 2.8e122

   1. Initial program 88.0%

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

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 2.8e122 < t

   1. Initial program 51.7%

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

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

   8. Applied rewrites 89.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 87.8% 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 2024228 
(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))))