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

Percentage Accurate: 77.2% → 88.9%

Time: 9.4s

Alternatives: 12

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: 77.2% 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: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2500:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{t - a}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{a - z}{t} \cdot y\right) \cdot \left(1 + \frac{a}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2500.0)
   (fma (/ y t) (- z a) x)
   (if (<= t 7.6e+147)
     (fma (- z t) (/ y (- t a)) (+ y x))
     (- x (* (* (/ (- a z) t) y) (+ 1.0 (/ a t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2500.0) {
		tmp = fma((y / t), (z - a), x);
	} else if (t <= 7.6e+147) {
		tmp = fma((z - t), (y / (t - a)), (y + x));
	} else {
		tmp = x - ((((a - z) / t) * y) * (1.0 + (a / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2500.0)
		tmp = fma(Float64(y / t), Float64(z - a), x);
	elseif (t <= 7.6e+147)
		tmp = fma(Float64(z - t), Float64(y / Float64(t - a)), Float64(y + x));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a - z) / t) * y) * Float64(1.0 + Float64(a / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2500.0], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 7.6e+147], N[(N[(z - t), $MachinePrecision] * N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision] * N[(1.0 + N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2500

   1. Initial program 56.3%

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

   3. Taylor expanded in t around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if -2500 < t < 7.59999999999999941e147

   1. Initial program 92.2%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 95.0

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

      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 95.0

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

   4. Applied rewrites 95.0%

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

   if 7.59999999999999941e147 < t

   1. Initial program 61.8%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 97.2%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2500:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{t - a}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{a - z}{t} \cdot y\right) \cdot \left(1 + \frac{a}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{if}\;t \leq -2500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.82 \cdot 10^{+109}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- z a) x)))
   (if (<= t -2500.0)
     t_1
     (if (<= t 1.82e+109) (- (+ 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((y / t), (z - a), x);
	double tmp;
	if (t <= -2500.0) {
		tmp = t_1;
	} else if (t <= 1.82e+109) {
		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(y / t), Float64(z - a), x)
	tmp = 0.0
	if (t <= -2500.0)
		tmp = t_1;
	elseif (t <= 1.82e+109)
		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[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -2500.0], t$95$1, If[LessEqual[t, 1.82e+109], 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 < -2500 or 1.82e109 < t

   1. Initial program 58.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   if -2500 < t < 1.82e109

   1. Initial program 93.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2500:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 1.82 \cdot 10^{+109}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{if}\;t \leq -2500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{t - a}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- z a) x)))
   (if (<= t -2500.0)
     t_1
     (if (<= t 1.85e+109) (fma (- z t) (/ y (- t a)) (+ y 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 <= -2500.0) {
		tmp = t_1;
	} else if (t <= 1.85e+109) {
		tmp = fma((z - t), (y / (t - a)), (y + 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 <= -2500.0)
		tmp = t_1;
	elseif (t <= 1.85e+109)
		tmp = fma(Float64(z - t), Float64(y / Float64(t - a)), Float64(y + 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, -2500.0], t$95$1, If[LessEqual[t, 1.85e+109], N[(N[(z - t), $MachinePrecision] * N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2500 or 1.8500000000000001e109 < t

   1. Initial program 58.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   if -2500 < t < 1.8500000000000001e109

   1. Initial program 93.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 95.4

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

      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 95.4

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

   4. Applied rewrites 95.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2500:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{t - a}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{if}\;t \leq -2500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.82 \cdot 10^{+109}:\\ \;\;\;\;\left(y + x\right) - \frac{z \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- z a) x)))
   (if (<= t -2500.0)
     t_1
     (if (<= t 1.82e+109) (- (+ 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((y / t), (z - a), x);
	double tmp;
	if (t <= -2500.0) {
		tmp = t_1;
	} else if (t <= 1.82e+109) {
		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(y / t), Float64(z - a), x)
	tmp = 0.0
	if (t <= -2500.0)
		tmp = t_1;
	elseif (t <= 1.82e+109)
		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[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -2500.0], t$95$1, If[LessEqual[t, 1.82e+109], 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 < -2500 or 1.82e109 < t

   1. Initial program 58.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   if -2500 < t < 1.82e109

   1. Initial program 93.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 93.0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2500:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 1.82 \cdot 10^{+109}:\\ \;\;\;\;\left(y + x\right) - \frac{z \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00145:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-59}:\\ \;\;\;\;\left(y + x\right) - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -0.00145)
   (fma (/ y t) (- z a) x)
   (if (<= t 1.8e-59) (- (+ y x) (/ (* z y) a)) (fma (/ z t) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -0.00145) {
		tmp = fma((y / t), (z - a), x);
	} else if (t <= 1.8e-59) {
		tmp = (y + x) - ((z * y) / a);
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -0.00145)
		tmp = fma(Float64(y / t), Float64(z - a), x);
	elseif (t <= 1.8e-59)
		tmp = Float64(Float64(y + x) - Float64(Float64(z * y) / a));
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -0.00145], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.8e-59], N[(N[(y + x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.00145

   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

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

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

   if -0.00145 < t < 1.8e-59

   1. Initial program 96.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{y \cdot z}{a}} \]
   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 88.8

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

   5. Applied rewrites 88.8%

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

   if 1.8e-59 < t

   1. Initial program 69.2%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r/ N/A

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower--.f64 N/A

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

      14. div-sub N/A

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

      15. associate-*r/ N/A

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

      16. associate-*r/ N/A

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

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

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

   8. Applied rewrites 71.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 84.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00145:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-59}:\\ \;\;\;\;\left(y + x\right) - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00145:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -0.00145)
   (fma (/ y t) (- z a) x)
   (if (<= t 1.8e-59) (fma y (- 1.0 (/ z a)) x) (fma (/ z t) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -0.00145) {
		tmp = fma((y / t), (z - a), x);
	} else if (t <= 1.8e-59) {
		tmp = fma(y, (1.0 - (z / a)), x);
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -0.00145)
		tmp = fma(Float64(y / t), Float64(z - a), x);
	elseif (t <= 1.8e-59)
		tmp = fma(y, Float64(1.0 - Float64(z / a)), x);
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -0.00145], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.8e-59], N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.00145

   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

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

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

   if -0.00145 < t < 1.8e-59

   1. Initial program 96.2%

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

   3. Taylor expanded in t around 0

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

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

   5. Applied rewrites 87.9%

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

   if 1.8e-59 < t

   1. Initial program 69.2%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r/ N/A

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower--.f64 N/A

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

      14. div-sub N/A

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

      15. associate-*r/ N/A

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

      16. associate-*r/ N/A

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

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

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

   8. Applied rewrites 71.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 84.6%

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

Alternative 7: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -3.9 \cdot 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -3.9e-21) t_1 (if (<= a 6.5e-123) (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 <= -3.9e-21) {
		tmp = t_1;
	} else if (a <= 6.5e-123) {
		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 <= -3.9e-21)
		tmp = t_1;
	elseif (a <= 6.5e-123)
		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, -3.9e-21], t$95$1, If[LessEqual[a, 6.5e-123], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.9000000000000001e-21 or 6.49999999999999938e-123 < a

   1. Initial program 83.0%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 84.3

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

   5. Applied rewrites 84.3%

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

   if -3.9000000000000001e-21 < a < 6.49999999999999938e-123

   1. Initial program 68.8%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 42.6

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r/ N/A

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower--.f64 N/A

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

      14. div-sub N/A

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

      15. associate-*r/ N/A

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

      16. associate-*r/ N/A

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

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

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 88.2%

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

Alternative 8: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-21}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -6.5e-21) (+ y x) (if (<= a 1.8e-48) (fma (/ z t) y x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -6.5e-21) {
		tmp = y + x;
	} else if (a <= 1.8e-48) {
		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 <= -6.5e-21)
		tmp = Float64(y + x);
	elseif (a <= 1.8e-48)
		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, -6.5e-21], N[(y + x), $MachinePrecision], If[LessEqual[a, 1.8e-48], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.49999999999999987e-21 or 1.8000000000000001e-48 < a

   1. Initial program 82.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

   if -6.49999999999999987e-21 < a < 1.8000000000000001e-48

   1. Initial program 70.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r/ N/A

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower--.f64 N/A

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

      14. div-sub N/A

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

      15. associate-*r/ N/A

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

      16. associate-*r/ N/A

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

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

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

   8. Applied rewrites 83.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 86.3%

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

Alternative 9: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-21}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -6.5e-21) (+ y x) (if (<= a 1.8e-48) (fma z (/ y t) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -6.5e-21) {
		tmp = y + x;
	} else if (a <= 1.8e-48) {
		tmp = fma(z, (y / t), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -6.5e-21)
		tmp = Float64(y + x);
	elseif (a <= 1.8e-48)
		tmp = fma(z, Float64(y / t), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -6.5e-21], N[(y + x), $MachinePrecision], If[LessEqual[a, 1.8e-48], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.49999999999999987e-21 or 1.8000000000000001e-48 < a

   1. Initial program 82.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

   if -6.49999999999999987e-21 < a < 1.8000000000000001e-48

   1. Initial program 70.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r/ N/A

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower--.f64 N/A

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

      14. div-sub N/A

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

      15. associate-*r/ N/A

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

      16. associate-*r/ N/A

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

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

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

   8. Applied rewrites 83.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 86.3%

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

   13. Applied rewrites 85.3%

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

Alternative 10: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{-209}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-307}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.35e-209)
   (+ y x)
   (if (<= a -1.15e-307) (* (/ y t) z) (if (<= a 8.8e-18) x (+ y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.35e-209) {
		tmp = y + x;
	} else if (a <= -1.15e-307) {
		tmp = (y / t) * z;
	} else if (a <= 8.8e-18) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-2.35d-209)) then
        tmp = y + x
    else if (a <= (-1.15d-307)) then
        tmp = (y / t) * z
    else if (a <= 8.8d-18) then
        tmp = x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.35e-209) {
		tmp = y + x;
	} else if (a <= -1.15e-307) {
		tmp = (y / t) * z;
	} else if (a <= 8.8e-18) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -2.35e-209:
		tmp = y + x
	elif a <= -1.15e-307:
		tmp = (y / t) * z
	elif a <= 8.8e-18:
		tmp = x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.35e-209)
		tmp = Float64(y + x);
	elseif (a <= -1.15e-307)
		tmp = Float64(Float64(y / t) * z);
	elseif (a <= 8.8e-18)
		tmp = x;
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -2.35e-209)
		tmp = y + x;
	elseif (a <= -1.15e-307)
		tmp = (y / t) * z;
	elseif (a <= 8.8e-18)
		tmp = x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.35e-209], N[(y + x), $MachinePrecision], If[LessEqual[a, -1.15e-307], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[a, 8.8e-18], x, N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.35e-209 or 8.7999999999999994e-18 < a

   1. Initial program 82.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   if -2.35e-209 < a < -1.1499999999999999e-307

   1. Initial program 75.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 26.5

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

   5. Applied rewrites 26.5%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r/ N/A

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower--.f64 N/A

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

      14. div-sub N/A

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

      15. associate-*r/ N/A

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

      16. associate-*r/ N/A

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

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

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

   8. Applied rewrites 87.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 71.2%

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

   if -1.1499999999999999e-307 < a < 8.7999999999999994e-18

   1. Initial program 64.8%

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

      1. lift--.f64 N/A

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

      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 68.9

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

      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 68.9

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

   4. Applied rewrites 68.9%

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

   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 69.9

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

   7. Applied rewrites 69.9%

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

   9. Applied rewrites 69.9%

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

Alternative 11: 62.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{-11}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.05e-11) (+ y x) (if (<= a 8.8e-18) x (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.05e-11) {
		tmp = y + x;
	} else if (a <= 8.8e-18) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-2.05d-11)) then
        tmp = y + x
    else if (a <= 8.8d-18) then
        tmp = x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.05e-11) {
		tmp = y + x;
	} else if (a <= 8.8e-18) {
		tmp = x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -2.05e-11:
		tmp = y + x
	elif a <= 8.8e-18:
		tmp = x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.05e-11)
		tmp = Float64(y + x);
	elseif (a <= 8.8e-18)
		tmp = x;
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -2.05e-11)
		tmp = y + x;
	elseif (a <= 8.8e-18)
		tmp = x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.05e-11], N[(y + x), $MachinePrecision], If[LessEqual[a, 8.8e-18], x, N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.05e-11 or 8.7999999999999994e-18 < a

   1. Initial program 83.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -2.05e-11 < a < 8.7999999999999994e-18

   1. Initial program 70.5%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 75.0

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

      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 75.0

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

   4. Applied rewrites 75.0%

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

   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 58.5

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

   7. Applied rewrites 58.5%

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

   9. Applied rewrites 58.5%

      \[\leadsto x \]
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 78.0%

   \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
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 83.2

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

   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 83.2

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

4. Applied rewrites 83.2%

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

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 55.7

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

7. Applied rewrites 55.7%

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

9. Applied rewrites 55.7%

   \[\leadsto x \]
10. 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 2024276 
(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))))