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

Percentage Accurate: 76.9% → 90.3%

Time: 8.7s

Alternatives: 10

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: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;\left(y + x\right) - \frac{-1}{\frac{\frac{t - a}{y}}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.05e+78)
   (fma (/ (- z a) t) y x)
   (if (<= t 1.55e+60)
     (- (+ y x) (/ -1.0 (/ (/ (- t a) y) (- z t))))
     (fma (/ y t) (- z a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.05e+78) {
		tmp = fma(((z - a) / t), y, x);
	} else if (t <= 1.55e+60) {
		tmp = (y + x) - (-1.0 / (((t - a) / y) / (z - t)));
	} else {
		tmp = fma((y / t), (z - a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.05e+78)
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	elseif (t <= 1.55e+60)
		tmp = Float64(Float64(y + x) - Float64(-1.0 / Float64(Float64(Float64(t - a) / y) / Float64(z - t))));
	else
		tmp = fma(Float64(y / t), Float64(z - a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.05e+78], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 1.55e+60], N[(N[(y + x), $MachinePrecision] - N[(-1.0 / N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.05e78

   1. Initial program 59.9%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-commutative 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. 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. associate-/l* N/A

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

      13. *-commutative 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 93.5

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

   8. Applied rewrites 93.5%

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

   if -1.05e78 < t < 1.55e60

   1. Initial program 91.8%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 94.8

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

   4. Applied rewrites 94.8%

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

   if 1.55e60 < t

   1. Initial program 57.8%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;\left(y + x\right) - \frac{-1}{\frac{\frac{t - a}{y}}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t}\\ t_2 := \left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-21}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-219}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_2 \leq 10^{+301}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* z y) t)) (t_2 (- (+ y x) (/ (* (- z t) y) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -2e-21)
       (+ y x)
       (if (<= t_2 4e-219) x (if (<= t_2 1e+301) (+ y x) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * y) / t;
	double t_2 = (y + x) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -2e-21) {
		tmp = y + x;
	} else if (t_2 <= 4e-219) {
		tmp = x;
	} else if (t_2 <= 1e+301) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * y) / t;
	double t_2 = (y + x) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -2e-21) {
		tmp = y + x;
	} else if (t_2 <= 4e-219) {
		tmp = x;
	} else if (t_2 <= 1e+301) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * y) / t
	t_2 = (y + x) - (((z - t) * y) / (a - t))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -2e-21:
		tmp = y + x
	elif t_2 <= 4e-219:
		tmp = x
	elif t_2 <= 1e+301:
		tmp = y + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * y) / t)
	t_2 = Float64(Float64(y + x) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -2e-21)
		tmp = Float64(y + x);
	elseif (t_2 <= 4e-219)
		tmp = x;
	elseif (t_2 <= 1e+301)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * y) / t;
	t_2 = (y + x) - (((z - t) * y) / (a - t));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -2e-21)
		tmp = y + x;
	elseif (t_2 <= 4e-219)
		tmp = x;
	elseif (t_2 <= 1e+301)
		tmp = y + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -2e-21], N[(y + x), $MachinePrecision], If[LessEqual[t$95$2, 4e-219], x, If[LessEqual[t$95$2, 1e+301], N[(y + x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0 or 1.00000000000000005e301 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 41.5%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 44.2%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -1.99999999999999982e-21 or 4.0000000000000001e-219 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 1.00000000000000005e301

   1. Initial program 97.4%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 73.4

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

   5. Applied rewrites 73.4%

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

   if -1.99999999999999982e-21 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 4.0000000000000001e-219

   1. Initial program 53.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 51.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 51.8

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot y + x \]

      4. mul0-lft N/A

         \[\leadsto \color{blue}{0} + x \]

      5. lower-+.f64 64.6

         \[\leadsto \color{blue}{0 + x} \]

   7. Applied rewrites 64.6%

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

   9. Applied rewrites 64.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq -\infty:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq -2 \cdot 10^{-21}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq 4 \cdot 10^{-219}:\\ \;\;\;\;x\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \leq 10^{+301}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{t - a}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -6e+73)
   (fma (/ (- z a) t) y x)
   (if (<= t 1.55e+60)
     (fma (- z t) (/ y (- t a)) (+ y x))
     (fma (/ y t) (- z a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -6e+73) {
		tmp = fma(((z - a) / t), y, x);
	} else if (t <= 1.55e+60) {
		tmp = fma((z - t), (y / (t - a)), (y + x));
	} else {
		tmp = fma((y / t), (z - a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -6e+73)
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	elseif (t <= 1.55e+60)
		tmp = fma(Float64(z - t), Float64(y / Float64(t - a)), Float64(y + x));
	else
		tmp = fma(Float64(y / t), Float64(z - a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -6e+73], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 1.55e+60], N[(N[(z - t), $MachinePrecision] * N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.00000000000000021e73

   1. Initial program 62.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-commutative 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. 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. associate-/l* N/A

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

      13. *-commutative 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 93.8

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

   8. Applied rewrites 93.8%

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

   if -6.00000000000000021e73 < t < 1.55e60

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 94.1

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 94.1

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

   4. Applied rewrites 94.1%

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

   if 1.55e60 < t

   1. Initial program 57.8%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{t - a}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+58}:\\ \;\;\;\;\left(y + x\right) - \frac{z}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.05e+78)
   (fma (/ (- z a) t) y x)
   (if (<= t 2.2e+58)
     (- (+ y x) (* (/ z (- a t)) y))
     (fma (/ y t) (- z a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.05e+78) {
		tmp = fma(((z - a) / t), y, x);
	} else if (t <= 2.2e+58) {
		tmp = (y + x) - ((z / (a - t)) * y);
	} else {
		tmp = fma((y / t), (z - a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.05e+78)
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	elseif (t <= 2.2e+58)
		tmp = Float64(Float64(y + x) - Float64(Float64(z / Float64(a - t)) * y));
	else
		tmp = fma(Float64(y / t), Float64(z - a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.05e+78], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 2.2e+58], N[(N[(y + x), $MachinePrecision] - N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.05e78

   1. Initial program 59.9%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-commutative 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. 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. associate-/l* N/A

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

      13. *-commutative 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 93.5

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

   8. Applied rewrites 93.5%

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

   if -1.05e78 < t < 2.2000000000000001e58

   1. Initial program 91.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 92.0

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

   8. Applied rewrites 92.0%

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

   if 2.2000000000000001e58 < t

   1. Initial program 57.8%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+58}:\\ \;\;\;\;\left(y + x\right) - \frac{z}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - a}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.2e+73)
   (fma (/ (- z a) t) y x)
   (if (<= t 4.6e-76) (fma y (- 1.0 (/ z a)) x) (fma (/ y t) (- z a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.2e+73) {
		tmp = fma(((z - a) / t), y, x);
	} else if (t <= 4.6e-76) {
		tmp = fma(y, (1.0 - (z / a)), x);
	} else {
		tmp = fma((y / t), (z - a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.2e+73)
		tmp = fma(Float64(Float64(z - a) / t), y, x);
	elseif (t <= 4.6e-76)
		tmp = fma(y, Float64(1.0 - Float64(z / a)), x);
	else
		tmp = fma(Float64(y / t), Float64(z - a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.2e+73], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t, 4.6e-76], N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.2000000000000003e73

   1. Initial program 62.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-commutative 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. 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. associate-/l* N/A

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

      13. *-commutative 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 93.8

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

   8. Applied rewrites 93.8%

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

   if -4.2000000000000003e73 < t < 4.60000000000000012e-76

   1. Initial program 92.4%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   if 4.60000000000000012e-76 < t

   1. Initial program 66.2%

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

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

   5. Applied rewrites 85.0%

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

Alternative 6: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- z a) x)))
   (if (<= t -4.2e+73) t_1 (if (<= t 4.6e-76) (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 <= -4.2e+73) {
		tmp = t_1;
	} else if (t <= 4.6e-76) {
		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 <= -4.2e+73)
		tmp = t_1;
	elseif (t <= 4.6e-76)
		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, -4.2e+73], t$95$1, If[LessEqual[t, 4.6e-76], 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 < -4.2000000000000003e73 or 4.60000000000000012e-76 < t

   1. Initial program 64.5%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 87.6

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

   5. Applied rewrites 87.6%

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

   if -4.2000000000000003e73 < t < 4.60000000000000012e-76

   1. Initial program 92.4%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

Alternative 7: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -4.5 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -4.5e-43) t_1 (if (<= a 2e-29) (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 <= -4.5e-43) {
		tmp = t_1;
	} else if (a <= 2e-29) {
		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 <= -4.5e-43)
		tmp = t_1;
	elseif (a <= 2e-29)
		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, -4.5e-43], t$95$1, If[LessEqual[a, 2e-29], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.50000000000000025e-43 or 1.99999999999999989e-29 < a

   1. Initial program 82.4%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 85.3

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

   5. Applied rewrites 85.3%

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

   if -4.50000000000000025e-43 < a < 1.99999999999999989e-29

   1. Initial program 73.2%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 70.7

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

   5. Applied rewrites 70.7%

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

   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 79.6%

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

Alternative 8: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-39}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.2e-39) (+ y x) (if (<= a 3.1e+26) (fma (/ z t) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.2e-39) {
		tmp = y + x;
	} else if (a <= 3.1e+26) {
		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 <= -1.2e-39)
		tmp = Float64(y + x);
	elseif (a <= 3.1e+26)
		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, -1.2e-39], N[(y + x), $MachinePrecision], If[LessEqual[a, 3.1e+26], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.20000000000000008e-39 or 3.1e26 < a

   1. Initial program 81.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 74.2

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

   5. Applied rewrites 74.2%

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

   if -1.20000000000000008e-39 < a < 3.1e26

   1. Initial program 74.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 71.9

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

   5. Applied rewrites 71.9%

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

   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 80.1%

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

Alternative 9: 62.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{+57}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.12e+82) x (if (<= t 1.18e+57) (+ y x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.12e+82) {
		tmp = x;
	} else if (t <= 1.18e+57) {
		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 <= (-1.12d+82)) then
        tmp = x
    else if (t <= 1.18d+57) 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 <= -1.12e+82) {
		tmp = x;
	} else if (t <= 1.18e+57) {
		tmp = y + x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.12e+82:
		tmp = x
	elif t <= 1.18e+57:
		tmp = y + x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.12e+82)
		tmp = x;
	elseif (t <= 1.18e+57)
		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 <= -1.12e+82)
		tmp = x;
	elseif (t <= 1.18e+57)
		tmp = y + x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.12e+82], x, If[LessEqual[t, 1.18e+57], N[(y + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -1.11999999999999998e82 or 1.18e57 < t

   1. Initial program 59.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 66.5

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 66.5

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

   4. Applied rewrites 66.5%

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

   5. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot y + x \]

      4. mul0-lft N/A

         \[\leadsto \color{blue}{0} + x \]

      5. lower-+.f64 65.9

         \[\leadsto \color{blue}{0 + x} \]

   7. Applied rewrites 65.9%

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

   9. Applied rewrites 65.9%

      \[\leadsto x \]

   if -1.11999999999999998e82 < t < 1.18e57

   1. Initial program 91.2%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 59.6

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

   5. Applied rewrites 59.6%

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

Alternative 10: 51.0% 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 78.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 82.6

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 82.6

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

4. Applied rewrites 82.6%

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

5. Taylor expanded in t around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot y + x \]

   4. mul0-lft N/A

      \[\leadsto \color{blue}{0} + x \]

   5. lower-+.f64 48.9

      \[\leadsto \color{blue}{0 + x} \]

7. Applied rewrites 48.9%

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

9. Applied rewrites 48.9%

   \[\leadsto x \]
10. Add Preprocessing

Developer Target 1: 87.9% 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 2024268 
(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))))