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

Percentage Accurate: 77.3% → 88.7%

Time: 8.9s

Alternatives: 13

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: 77.3% 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: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{if}\;t \leq -1.46 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\left(y + x\right) - \frac{-1}{\frac{\frac{t - a}{y}}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z a) t) y x)))
   (if (<= t -1.46e+35)
     t_1
     (if (<= t 0.8) (- (+ y x) (/ -1.0 (/ (/ (- t a) y) (- z t)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - a) / t), y, x);
	double tmp;
	if (t <= -1.46e+35) {
		tmp = t_1;
	} else if (t <= 0.8) {
		tmp = (y + x) - (-1.0 / (((t - a) / y) / (z - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - a) / t), y, x)
	tmp = 0.0
	if (t <= -1.46e+35)
		tmp = t_1;
	elseif (t <= 0.8)
		tmp = Float64(Float64(y + x) - Float64(-1.0 / Float64(Float64(Float64(t - a) / y) / Float64(z - t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -1.46e+35], t$95$1, If[LessEqual[t, 0.8], N[(N[(y + x), $MachinePrecision] - N[(-1.0 / N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.4599999999999999e35 or 0.80000000000000004 < t

   1. Initial program 62.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{a \cdot y}{t}\right) - -1 \cdot \frac{y \cdot z}{t}} \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

   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. 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-+l+ 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. unsub-neg N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 90.9

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

   8. Applied rewrites 90.9%

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

   if -1.4599999999999999e35 < t < 0.80000000000000004

   1. Initial program 92.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 93.4

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

   4. Applied rewrites 93.4%

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{1}{\frac{\frac{a - t}{y}}{z - t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\left(y + x\right) - \frac{-1}{\frac{\frac{t - a}{y}}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-174}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t\_1 \leq 10^{-187}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ y x) (/ (* (- z t) y) (- a t)))))
   (if (<= t_1 (- INFINITY))
     (/ (* z y) t)
     (if (<= t_1 -2e-174) (+ y x) (if (<= t_1 1e-187) x (+ y x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (z * y) / t;
	} else if (t_1 <= -2e-174) {
		tmp = y + x;
	} else if (t_1 <= 1e-187) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (z * y) / t;
	} else if (t_1 <= -2e-174) {
		tmp = y + x;
	} else if (t_1 <= 1e-187) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y + x) - (((z - t) * y) / (a - t))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (z * y) / t
	elif t_1 <= -2e-174:
		tmp = y + x
	elif t_1 <= 1e-187:
		tmp = x
	else:
		tmp = y + x
	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)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(z * y) / t);
	elseif (t_1 <= -2e-174)
		tmp = Float64(y + x);
	elseif (t_1 <= 1e-187)
		tmp = x;
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y + x) - (((z - t) * y) / (a - t));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (z * y) / t;
	elseif (t_1 <= -2e-174)
		tmp = y + x;
	elseif (t_1 <= 1e-187)
		tmp = x;
	else
		tmp = y + x;
	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] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, -2e-174], N[(y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-187], x, N[(y + x), $MachinePrecision]]]]]

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 16.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 36.7%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -2e-174 or 1e-187 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 89.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 66.0

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

   5. Applied rewrites 66.0%

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

   if -2e-174 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 1e-187

   1. Initial program 30.3%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. div-sub N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 57.1%

      \[\leadsto 0 + \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.1%

       \[\leadsto x \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq -\infty:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq -2 \cdot 10^{-174}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq 10^{-187}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.22 \cdot 10^{+46}:\\ \;\;\;\;x - \left(\left(a - z\right) \cdot \frac{y}{t}\right) \cdot \left(\frac{a}{t} + 1\right)\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.22e+46)
   (- x (* (* (- a z) (/ y t)) (+ (/ a t) 1.0)))
   (if (<= t 0.8)
     (- (+ y x) (/ (* (- z t) y) (- a t)))
     (fma (/ (- z a) t) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.22e+46) {
		tmp = x - (((a - z) * (y / t)) * ((a / t) + 1.0));
	} else if (t <= 0.8) {
		tmp = (y + x) - (((z - t) * y) / (a - t));
	} else {
		tmp = fma(((z - a) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.22e+46)
		tmp = Float64(x - Float64(Float64(Float64(a - z) * Float64(y / t)) * Float64(Float64(a / t) + 1.0)));
	elseif (t <= 0.8)
		tmp = Float64(Float64(y + x) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)));
	else
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.22e+46], N[(x - N[(N[(N[(a - z), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision] * N[(N[(a / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.8], N[(N[(y + x), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.21999999999999999e46

   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 90.3%

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

   if -2.21999999999999999e46 < t < 0.80000000000000004

   1. Initial program 92.6%

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

   if 0.80000000000000004 < t

   1. Initial program 63.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 88.3

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

   5. Applied rewrites 88.3%

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

   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. 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-+l+ 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. unsub-neg N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 91.1

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

   8. Applied rewrites 91.1%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.22 \cdot 10^{+46}:\\ \;\;\;\;x - \left(\left(a - z\right) \cdot \frac{y}{t}\right) \cdot \left(\frac{a}{t} + 1\right)\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2.22 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z a) t) y x)))
   (if (<= t -2.22e+46)
     t_1
     (if (<= t 0.8) (- (+ y x) (/ (* (- z t) y) (- a t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - a) / t), y, x);
	double tmp;
	if (t <= -2.22e+46) {
		tmp = t_1;
	} else if (t <= 0.8) {
		tmp = (y + x) - (((z - t) * y) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - a) / t), y, x)
	tmp = 0.0
	if (t <= -2.22e+46)
		tmp = t_1;
	elseif (t <= 0.8)
		tmp = Float64(Float64(y + x) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.22e+46], t$95$1, If[LessEqual[t, 0.8], N[(N[(y + x), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.21999999999999999e46 or 0.80000000000000004 < t

   1. Initial program 61.3%

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

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

   5. Applied rewrites 89.0%

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

   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. 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-+l+ 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. unsub-neg N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 90.7

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

   8. Applied rewrites 90.7%

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

   if -2.21999999999999999e46 < t < 0.80000000000000004

   1. Initial program 92.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.22 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{if}\;t \leq -0.125:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.021:\\ \;\;\;\;\left(y + x\right) - \frac{z \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z a) t) y x)))
   (if (<= t -0.125)
     t_1
     (if (<= t 0.021) (- (+ y x) (/ (* z y) (- a t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - a) / t), y, x);
	double tmp;
	if (t <= -0.125) {
		tmp = t_1;
	} else if (t <= 0.021) {
		tmp = (y + x) - ((z * y) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - a) / t), y, x)
	tmp = 0.0
	if (t <= -0.125)
		tmp = t_1;
	elseif (t <= 0.021)
		tmp = Float64(Float64(y + x) - Float64(Float64(z * y) / Float64(a - t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -0.125], t$95$1, If[LessEqual[t, 0.021], N[(N[(y + x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.125 or 0.0210000000000000013 < t

   1. Initial program 63.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 87.7

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

   5. Applied rewrites 87.7%

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

   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. 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-+l+ 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. unsub-neg N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 89.5

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

   8. Applied rewrites 89.5%

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

   if -0.125 < t < 0.0210000000000000013

   1. Initial program 94.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.125:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 0.021:\\ \;\;\;\;\left(y + x\right) - \frac{z \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.6e-61)
   (fma (/ y t) (- z a) x)
   (if (<= t 2.8e-53)
     (fma y (- 1.0 (/ (- z t) 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 <= -7.6e-61) {
		tmp = fma((y / t), (z - a), x);
	} else if (t <= 2.8e-53) {
		tmp = fma(y, (1.0 - ((z - t) / 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 <= -7.6e-61)
		tmp = fma(Float64(y / t), Float64(z - a), x);
	elseif (t <= 2.8e-53)
		tmp = fma(y, Float64(1.0 - Float64(Float64(z - t) / 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, -7.6e-61], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 2.8e-53], N[(y * N[(1.0 - N[(N[(z - t), $MachinePrecision] / 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 < -7.59999999999999961e-61

   1. Initial program 67.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

   if -7.59999999999999961e-61 < t < 2.79999999999999985e-53

   1. Initial program 92.8%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if 2.79999999999999985e-53 < t

   1. Initial program 69.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 87.9

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

   5. Applied rewrites 87.9%

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

   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. 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-+l+ 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. unsub-neg N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 90.2

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

   8. Applied rewrites 90.2%

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

Alternative 7: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.6e-61)
   (fma (/ y t) (- z a) x)
   (if (<= t 2.8e-53) (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 <= -7.6e-61) {
		tmp = fma((y / t), (z - a), x);
	} else if (t <= 2.8e-53) {
		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 <= -7.6e-61)
		tmp = fma(Float64(y / t), Float64(z - a), x);
	elseif (t <= 2.8e-53)
		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, -7.6e-61], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 2.8e-53], 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 < -7.59999999999999961e-61

   1. Initial program 67.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

   if -7.59999999999999961e-61 < t < 2.79999999999999985e-53

   1. Initial program 92.8%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 86.4

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

   5. Applied rewrites 86.4%

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

   if 2.79999999999999985e-53 < t

   1. Initial program 69.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 87.9

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

   5. Applied rewrites 87.9%

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

   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. 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-+l+ 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. unsub-neg N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 90.2

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

   8. Applied rewrites 90.2%

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

Alternative 8: 81.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- z a) x)))
   (if (<= t -7.6e-61) t_1 (if (<= t 2.8e-53) (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 <= -7.6e-61) {
		tmp = t_1;
	} else if (t <= 2.8e-53) {
		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 <= -7.6e-61)
		tmp = t_1;
	elseif (t <= 2.8e-53)
		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, -7.6e-61], t$95$1, If[LessEqual[t, 2.8e-53], 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 < -7.59999999999999961e-61 or 2.79999999999999985e-53 < t

   1. Initial program 68.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{a \cdot y}{t}\right) - -1 \cdot \frac{y \cdot z}{t}} \]
   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 86.5

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

   5. Applied rewrites 86.5%

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

   if -7.59999999999999961e-61 < t < 2.79999999999999985e-53

   1. Initial program 92.8%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 86.4

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

   5. Applied rewrites 86.4%

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

Alternative 9: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -3.5e-38) t_1 (if (<= a 2.8e-8) (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 <= -3.5e-38) {
		tmp = t_1;
	} else if (a <= 2.8e-8) {
		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 <= -3.5e-38)
		tmp = t_1;
	elseif (a <= 2.8e-8)
		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, -3.5e-38], t$95$1, If[LessEqual[a, 2.8e-8], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5000000000000001e-38 or 2.7999999999999999e-8 < a

   1. Initial program 77.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   if -3.5000000000000001e-38 < a < 2.7999999999999999e-8

   1. Initial program 76.3%

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

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 85.5%

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

Alternative 10: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-24}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 10.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.05e-24) (+ y x) (if (<= a 10.5) (fma (/ z t) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.05e-24) {
		tmp = y + x;
	} else if (a <= 10.5) {
		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 <= -1.05e-24)
		tmp = Float64(y + x);
	elseif (a <= 10.5)
		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, -1.05e-24], N[(y + x), $MachinePrecision], If[LessEqual[a, 10.5], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.05e-24 or 10.5 < a

   1. Initial program 78.0%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 70.9

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

   5. Applied rewrites 70.9%

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

   if -1.05e-24 < a < 10.5

   1. Initial program 75.9%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 83.9%

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

Alternative 11: 63.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+178}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+211}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.8e+178) x (if (<= t 3.8e+211) (+ y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.8e+178) {
		tmp = x;
	} else if (t <= 3.8e+211) {
		tmp = y + x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-2.8d+178)) then
        tmp = x
    else if (t <= 3.8d+211) then
        tmp = y + x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.8e+178) {
		tmp = x;
	} else if (t <= 3.8e+211) {
		tmp = y + x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -2.8e+178:
		tmp = x
	elif t <= 3.8e+211:
		tmp = y + x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.8e+178)
		tmp = x;
	elseif (t <= 3.8e+211)
		tmp = Float64(y + x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -2.8e+178)
		tmp = x;
	elseif (t <= 3.8e+211)
		tmp = y + x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.8e+178], x, If[LessEqual[t, 3.8e+211], N[(y + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -2.79999999999999993e178 or 3.80000000000000016e211 < t

   1. Initial program 49.4%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l/ N/A

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

      4. div-sub N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 70.2%

      \[\leadsto 0 + \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.2%

       \[\leadsto x \]

   if -2.79999999999999993e178 < t < 3.80000000000000016e211

   1. Initial program 85.2%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 57.3

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

   5. Applied rewrites 57.3%

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

Alternative 12: 50.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-/l/ N/A

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

   4. div-sub N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 82.8%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 44.9%

   \[\leadsto 0 + \color{blue}{x} \]
9. Step-by-step derivation

10. Applied rewrites 44.9%

    \[\leadsto x \]
11. Add Preprocessing

Alternative 13: 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 76.9%

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

3. Taylor expanded in z around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 57.6

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

5. Applied rewrites 57.6%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 21.0%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 2.7%

    \[\leadsto 0 \]
12. Add Preprocessing

Developer Target 1: 88.2% 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 2024254 
(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))))