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

Percentage Accurate: 76.6% → 91.1%

Time: 7.7s

Alternatives: 8

Speedup: 0.9×

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: 76.6% 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: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-288}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ y x) (/ (* (- z t) y) (- a t))))
        (t_2 (fma (/ (- z a) t) y x)))
   (if (<= t_1 (- INFINITY))
     (fma (/ y t) (- z a) x)
     (if (<= t_1 -5e-237)
       t_1
       (if (<= t_1 2e-288) t_2 (if (<= t_1 2e+303) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * y) / (a - t));
	double t_2 = fma(((z - a) / t), y, x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y / t), (z - a), x);
	} else if (t_1 <= -5e-237) {
		tmp = t_1;
	} else if (t_1 <= 2e-288) {
		tmp = t_2;
	} else if (t_1 <= 2e+303) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y + x) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	t_2 = fma(Float64(Float64(z - a) / t), y, x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(y / t), Float64(z - a), x);
	elseif (t_1 <= -5e-237)
		tmp = t_1;
	elseif (t_1 <= 2e-288)
		tmp = t_2;
	elseif (t_1 <= 2e+303)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, -5e-237], t$95$1, If[LessEqual[t$95$1, 2e-288], t$95$2, If[LessEqual[t$95$1, 2e+303], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0

   1. Initial program 29.8%

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

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

   5. Applied rewrites 74.3%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -5.0000000000000002e-237 or 2.00000000000000012e-288 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 2e303

   1. Initial program 98.4%

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

   if -5.0000000000000002e-237 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 2.00000000000000012e-288 or 2e303 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 20.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 23.0%

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

   9. Taylor expanded in t around inf

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

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

   11. Applied rewrites 80.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq -5 \cdot 10^{-237}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq 2 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.5e+68)
   (fma (/ y t) (- z a) x)
   (if (<= t 2.2e+37) (fma y (- 1.0 (/ z a)) x) (fma (/ (- z a) t) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.5e+68) {
		tmp = fma((y / t), (z - a), x);
	} else if (t <= 2.2e+37) {
		tmp = fma(y, (1.0 - (z / a)), x);
	} else {
		tmp = fma(((z - a) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.5e+68)
		tmp = fma(Float64(y / t), Float64(z - a), x);
	elseif (t <= 2.2e+37)
		tmp = fma(y, Float64(1.0 - Float64(z / a)), x);
	else
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.5e+68], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 2.2e+37], N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.5000000000000003e68

   1. Initial program 56.8%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   if -4.5000000000000003e68 < t < 2.2000000000000001e37

   1. Initial program 90.1%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if 2.2000000000000001e37 < t

   1. Initial program 55.8%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.9%

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

   9. Taylor expanded in t around inf

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

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

   11. Applied rewrites 84.1%

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

Alternative 3: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- z a) x)))
   (if (<= t -4.5e+68) t_1 (if (<= t 2.2e+37) (fma y (- 1.0 (/ z a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / t), (z - a), x);
	double tmp;
	if (t <= -4.5e+68) {
		tmp = t_1;
	} else if (t <= 2.2e+37) {
		tmp = fma(y, (1.0 - (z / a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / t), Float64(z - a), x)
	tmp = 0.0
	if (t <= -4.5e+68)
		tmp = t_1;
	elseif (t <= 2.2e+37)
		tmp = fma(y, Float64(1.0 - Float64(z / a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -4.5e+68], t$95$1, If[LessEqual[t, 2.2e+37], N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.5000000000000003e68 or 2.2000000000000001e37 < t

   1. Initial program 56.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 86.1

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

   5. Applied rewrites 86.1%

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

   if -4.5000000000000003e68 < t < 2.2000000000000001e37

   1. Initial program 90.1%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

Alternative 4: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{-98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -1.45e-98) t_1 (if (<= a 4e-11) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (z / a)), x);
	double tmp;
	if (a <= -1.45e-98) {
		tmp = t_1;
	} else if (a <= 4e-11) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(z / a)), x)
	tmp = 0.0
	if (a <= -1.45e-98)
		tmp = t_1;
	elseif (a <= 4e-11)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.45e-98], t$95$1, If[LessEqual[a, 4e-11], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e-98 or 3.99999999999999976e-11 < a

   1. Initial program 81.1%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -1.45e-98 < a < 3.99999999999999976e-11

   1. Initial program 68.4%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 83.1%

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

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-29}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -7e-29) (+ y x) (if (<= a 2e-42) (fma (/ z t) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7e-29) {
		tmp = y + x;
	} else if (a <= 2e-42) {
		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 <= -7e-29)
		tmp = Float64(y + x);
	elseif (a <= 2e-42)
		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, -7e-29], N[(y + x), $MachinePrecision], If[LessEqual[a, 2e-42], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.9999999999999995e-29 or 2.00000000000000008e-42 < a

   1. Initial program 79.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if -6.9999999999999995e-29 < a < 2.00000000000000008e-42

   1. Initial program 71.6%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 83.1%

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

Alternative 6: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.05 \cdot 10^{+57}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(-1 + 1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.05e+57)
   (+ y x)
   (if (<= a 8.4e-48) (fma (+ -1.0 1.0) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.05e+57) {
		tmp = y + x;
	} else if (a <= 8.4e-48) {
		tmp = fma((-1.0 + 1.0), y, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3.05e+57)
		tmp = Float64(y + x);
	elseif (a <= 8.4e-48)
		tmp = fma(Float64(-1.0 + 1.0), y, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3.05e+57], N[(y + x), $MachinePrecision], If[LessEqual[a, 8.4e-48], N[(N[(-1.0 + 1.0), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.04999999999999988e57 or 8.39999999999999954e-48 < a

   1. Initial program 78.7%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 81.2

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

   5. Applied rewrites 81.2%

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

   if -3.04999999999999988e57 < a < 8.39999999999999954e-48

   1. Initial program 74.0%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 57.6%

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

Alternative 7: 60.1% 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 76.5%

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

3. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 66.3

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

5. Applied rewrites 66.3%

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

Alternative 8: 2.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 76.5%

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

3. Taylor expanded in y around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

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

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

   5. associate-/l* N/A

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

   6. mul-1-neg N/A

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

   7. distribute-lft-in N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. remove-double-neg N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. associate--l+ N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

5. Applied rewrites 36.4%

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

6. Taylor expanded in t around inf

   \[\leadsto y + \color{blue}{-1 \cdot y} \]
7. Step-by-step derivation

8. Applied rewrites 2.6%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer Target 1: 88.3% 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 2024235 
(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))))