Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 97.3% → 99.7%

Time: 11.6s

Alternatives: 13

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

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 - z) + 1.0d0) / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y - z) / (((t - z) + 1.0) / a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

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 - z) + 1.0d0) / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y - z) / (((t - z) + 1.0) / a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y - z}{-1 + \left(z - t\right)}, a, x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma (/ (- y z) (+ -1.0 (- z t))) a x))

C

double code(double x, double y, double z, double t, double a) {
	return fma(((y - z) / (-1.0 + (z - t))), a, x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(Float64(y - z) / Float64(-1.0 + Float64(z - t))), a, x)
end

Wolfram

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

Derivation

1. Initial program 97.7%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-/r/ N/A

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

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

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

   5. distribute-frac-neg2 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. --lowering--.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-in N/A

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

   11. metadata-eval N/A

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

   12. unsub-neg N/A

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

   13. --lowering--.f64 N/A

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

   14. --lowering--.f64 99.2

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

   \[\leadsto \mathsf{fma}\left(\frac{y - z}{-1 + \left(z - t\right)}, a, x\right) \]
6. Add Preprocessing

Alternative 2: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{+188}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -46000:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.65e+188)
   (- x a)
   (if (<= z -46000.0)
     (fma a (/ y z) x)
     (if (<= z 3e+44)
       (fma a (/ y (- -1.0 t)) x)
       (if (<= z 2.4e+148) (fma (/ a z) y x) (- x a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e+188) {
		tmp = x - a;
	} else if (z <= -46000.0) {
		tmp = fma(a, (y / z), x);
	} else if (z <= 3e+44) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else if (z <= 2.4e+148) {
		tmp = fma((a / z), y, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.65e+188)
		tmp = Float64(x - a);
	elseif (z <= -46000.0)
		tmp = fma(a, Float64(y / z), x);
	elseif (z <= 3e+44)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	elseif (z <= 2.4e+148)
		tmp = fma(Float64(a / z), y, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.65e+188], N[(x - a), $MachinePrecision], If[LessEqual[z, -46000.0], N[(a * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3e+44], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.4e+148], N[(N[(a / z), $MachinePrecision] * y + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.64999999999999994e188 or 2.39999999999999995e148 < z

   1. Initial program 97.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 87.2%

      \[\leadsto x - \color{blue}{a} \]

   if -2.64999999999999994e188 < z < -46000

   1. Initial program 90.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. distribute-lft-neg-in N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 76.2%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 76.5

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

   10. Simplified 76.5%

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

   if -46000 < z < 2.99999999999999987e44

   1. Initial program 99.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 89.0

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

   5. Simplified 89.0%

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

   if 2.99999999999999987e44 < z < 2.39999999999999995e148

   1. Initial program 100.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. distribute-lft-neg-in N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 93.2%

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

   8. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 93.2

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

   10. Simplified 93.2%

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

Alternative 3: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{+188}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -0.82:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(-a, \mathsf{fma}\left(y, z, y\right), x\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.65e+188)
   (- x a)
   (if (<= z -0.82)
     (fma a (/ y z) x)
     (if (<= z 1e-18)
       (fma (- a) (fma y z y) x)
       (if (<= z 2.7e+148) (fma (/ a z) y x) (- x a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e+188) {
		tmp = x - a;
	} else if (z <= -0.82) {
		tmp = fma(a, (y / z), x);
	} else if (z <= 1e-18) {
		tmp = fma(-a, fma(y, z, y), x);
	} else if (z <= 2.7e+148) {
		tmp = fma((a / z), y, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.65e+188)
		tmp = Float64(x - a);
	elseif (z <= -0.82)
		tmp = fma(a, Float64(y / z), x);
	elseif (z <= 1e-18)
		tmp = fma(Float64(-a), fma(y, z, y), x);
	elseif (z <= 2.7e+148)
		tmp = fma(Float64(a / z), y, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.65e+188], N[(x - a), $MachinePrecision], If[LessEqual[z, -0.82], N[(a * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1e-18], N[((-a) * N[(y * z + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.7e+148], N[(N[(a / z), $MachinePrecision] * y + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.64999999999999994e188 or 2.70000000000000019e148 < z

   1. Initial program 97.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 87.2%

      \[\leadsto x - \color{blue}{a} \]

   if -2.64999999999999994e188 < z < -0.819999999999999951

   1. Initial program 90.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. distribute-lft-neg-in N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 76.2%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 76.5

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

   10. Simplified 76.5%

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

   if -0.819999999999999951 < z < 1.0000000000000001e-18

   1. Initial program 99.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 72.4

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

   5. Simplified 72.4%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 70.4%

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

   9. Taylor expanded in z around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. associate-+l+ N/A

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

       4. +-commutative N/A

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

       5. +-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. distribute-lft-out N/A

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

       9. accelerator-lowering-fma.f64 N/A

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

       10. mul-1-neg N/A

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

       11. neg-lowering-neg.f64 N/A

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

       12. +-commutative N/A

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

       13. accelerator-lowering-fma.f64 70.3

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

   11. Simplified 70.3%

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

   if 1.0000000000000001e-18 < z < 2.70000000000000019e148

   1. Initial program 97.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. distribute-lft-neg-in N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 83.6%

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

   8. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 73.0

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

   10. Simplified 73.0%

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

Alternative 4: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{y}{z}, x\right)\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+188}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -0.76:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.000106:\\ \;\;\;\;\mathsf{fma}\left(-a, \mathsf{fma}\left(y, z, y\right), x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ y z) x)))
   (if (<= z -2.7e+188)
     (- x a)
     (if (<= z -0.76)
       t_1
       (if (<= z 0.000106)
         (fma (- a) (fma y z y) x)
         (if (<= z 2.5e+148) t_1 (- x a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (y / z), x);
	double tmp;
	if (z <= -2.7e+188) {
		tmp = x - a;
	} else if (z <= -0.76) {
		tmp = t_1;
	} else if (z <= 0.000106) {
		tmp = fma(-a, fma(y, z, y), x);
	} else if (z <= 2.5e+148) {
		tmp = t_1;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(y / z), x)
	tmp = 0.0
	if (z <= -2.7e+188)
		tmp = Float64(x - a);
	elseif (z <= -0.76)
		tmp = t_1;
	elseif (z <= 0.000106)
		tmp = fma(Float64(-a), fma(y, z, y), x);
	elseif (z <= 2.5e+148)
		tmp = t_1;
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.7e+188], N[(x - a), $MachinePrecision], If[LessEqual[z, -0.76], t$95$1, If[LessEqual[z, 0.000106], N[((-a) * N[(y * z + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.5e+148], t$95$1, N[(x - a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7e188 or 2.50000000000000012e148 < z

   1. Initial program 97.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 87.2%

      \[\leadsto x - \color{blue}{a} \]

   if -2.7e188 < z < -0.76000000000000001 or 1.06e-4 < z < 2.50000000000000012e148

   1. Initial program 93.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. distribute-lft-neg-in N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 83.8%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 79.2

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

   10. Simplified 79.2%

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

   if -0.76000000000000001 < z < 1.06e-4

   1. Initial program 99.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 68.2%

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

   9. Taylor expanded in z around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. associate-+l+ N/A

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

       4. +-commutative N/A

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

       5. +-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. distribute-lft-out N/A

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

       9. accelerator-lowering-fma.f64 N/A

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

       10. mul-1-neg N/A

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

       11. neg-lowering-neg.f64 N/A

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

       12. +-commutative N/A

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

       13. accelerator-lowering-fma.f64 68.1

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

   11. Simplified 68.1%

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

Alternative 5: 54.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq -1.5 \cdot 10^{+59}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- y z) (/ (+ (- t z) 1.0) a)) -1.5e+59) (- a) x))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((y - z) / (((t - z) + 1.0) / a)) <= -1.5e+59) {
		tmp = -a;
	} 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 (((y - z) / (((t - z) + 1.0d0) / a)) <= (-1.5d+59)) then
        tmp = -a
    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 (((y - z) / (((t - z) + 1.0) / a)) <= -1.5e+59) {
		tmp = -a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((y - z) / (((t - z) + 1.0) / a)) <= -1.5e+59:
		tmp = -a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)) <= -1.5e+59)
		tmp = Float64(-a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((y - z) / (((t - z) + 1.0) / a)) <= -1.5e+59)
		tmp = -a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], -1.5e+59], (-a), x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -1.5e59

   1. Initial program 99.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 42.3%

      \[\leadsto x - \color{blue}{a} \]

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(a\right)} \]

      2. neg-lowering-neg.f64 32.2

         \[\leadsto \color{blue}{-a} \]

   8. Simplified 32.2%

      \[\leadsto \color{blue}{-a} \]

   if -1.5e59 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 97.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 65.7%

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

Alternative 6: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{-1 + \left(z - t\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - \left(z + -1\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.75e+67)
   (fma (/ (- y z) z) a x)
   (if (<= z 3.3e+148)
     (fma (/ a (+ -1.0 (- z t))) y x)
     (fma a (/ z (- t (+ z -1.0))) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+67) {
		tmp = fma(((y - z) / z), a, x);
	} else if (z <= 3.3e+148) {
		tmp = fma((a / (-1.0 + (z - t))), y, x);
	} else {
		tmp = fma(a, (z / (t - (z + -1.0))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.75e+67)
		tmp = fma(Float64(Float64(y - z) / z), a, x);
	elseif (z <= 3.3e+148)
		tmp = fma(Float64(a / Float64(-1.0 + Float64(z - t))), y, x);
	else
		tmp = fma(a, Float64(z / Float64(t - Float64(z + -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.75e+67], N[(N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 3.3e+148], N[(N[(a / N[(-1.0 + N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(a * N[(z / N[(t - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.75e67

   1. Initial program 94.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 94.4%

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

   if -1.75e67 < z < 3.3000000000000001e148

   1. Initial program 98.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. distribute-lft-neg-in N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 90.9%

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

   if 3.3000000000000001e148 < z

   1. Initial program 97.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{-1 + \left(z - t\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - \left(z + -1\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{a}{-t}, x\right)\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{z + -1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ a (- t)) x)))
   (if (<= t -3.2e+79)
     t_1
     (if (<= t 3.6e+126) (fma (- y z) (/ a (+ z -1.0)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (a / -t), x);
	double tmp;
	if (t <= -3.2e+79) {
		tmp = t_1;
	} else if (t <= 3.6e+126) {
		tmp = fma((y - z), (a / (z + -1.0)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(a / Float64(-t)), x)
	tmp = 0.0
	if (t <= -3.2e+79)
		tmp = t_1;
	elseif (t <= 3.6e+126)
		tmp = fma(Float64(y - z), Float64(a / Float64(z + -1.0)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(a / (-t)), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -3.2e+79], t$95$1, If[LessEqual[t, 3.6e+126], N[(N[(y - z), $MachinePrecision] * N[(a / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.20000000000000003e79 or 3.6e126 < t

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 79.1

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

   5. Simplified 79.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 89.5

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

   7. Applied egg-rr 89.5%

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

   if -3.20000000000000003e79 < t < 3.6e126

   1. Initial program 98.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. mul-1-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 94.2%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{-t}, x\right)\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{z + -1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{-t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \mathbf{if}\;z \leq -3.25 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) z) a x)))
   (if (<= z -3.25e+41)
     t_1
     (if (<= z 3.5e+44) (fma a (/ y (- -1.0 t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / z), a, x);
	double tmp;
	if (z <= -3.25e+41) {
		tmp = t_1;
	} else if (z <= 3.5e+44) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / z), a, x)
	tmp = 0.0
	if (z <= -3.25e+41)
		tmp = t_1;
	elseif (z <= 3.5e+44)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -3.25e+41], t$95$1, If[LessEqual[z, 3.5e+44], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.24999999999999988e41 or 3.4999999999999999e44 < z

   1. Initial program 95.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 93.0%

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

   if -3.24999999999999988e41 < z < 3.4999999999999999e44

   1. Initial program 99.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 88.7

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

   5. Simplified 88.7%

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

Alternative 9: 87.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a}{z}, y - z, x\right)\\ \mathbf{if}\;z \leq -3 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ a z) (- y z) x)))
   (if (<= z -3e+41) t_1 (if (<= z 2.7e+44) (fma a (/ y (- -1.0 t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((a / z), (y - z), x);
	double tmp;
	if (z <= -3e+41) {
		tmp = t_1;
	} else if (z <= 2.7e+44) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(a / z), Float64(y - z), x)
	tmp = 0.0
	if (z <= -3e+41)
		tmp = t_1;
	elseif (z <= 2.7e+44)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a / z), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -3e+41], t$95$1, If[LessEqual[z, 2.7e+44], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9999999999999998e41 or 2.7e44 < z

   1. Initial program 95.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. distribute-lft-neg-in N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 96.2%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 90.2

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

   7. Simplified 90.2%

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

   if -2.9999999999999998e41 < z < 2.7e44

   1. Initial program 99.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 88.7

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

   5. Simplified 88.7%

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

Alternative 10: 97.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{a}{-1 + \left(z - t\right)}, y - z, x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma (/ a (+ -1.0 (- z t))) (- y z) x))

C

double code(double x, double y, double z, double t, double a) {
	return fma((a / (-1.0 + (z - t))), (y - z), x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(a / Float64(-1.0 + Float64(z - t))), Float64(y - z), x)
end

Wolfram

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

Derivation

1. Initial program 97.7%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. clear-num N/A

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

   6. distribute-lft-neg-in N/A

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

   7. clear-num N/A

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

   8. distribute-frac-neg2 N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 97.9%

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

5. Final simplification 97.9%

   \[\leadsto \mathsf{fma}\left(\frac{a}{-1 + \left(z - t\right)}, y - z, x\right) \]
6. Add Preprocessing

Alternative 11: 73.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+41}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+22}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.7e+41) (- x a) (if (<= z 7.5e+22) (- x (* y a)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.7e+41) {
		tmp = x - a;
	} else if (z <= 7.5e+22) {
		tmp = x - (y * a);
	} else {
		tmp = x - a;
	}
	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 (z <= (-3.7d+41)) then
        tmp = x - a
    else if (z <= 7.5d+22) then
        tmp = x - (y * a)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.7e+41) {
		tmp = x - a;
	} else if (z <= 7.5e+22) {
		tmp = x - (y * a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.7e+41:
		tmp = x - a
	elif z <= 7.5e+22:
		tmp = x - (y * a)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.7e+41)
		tmp = Float64(x - a);
	elseif (z <= 7.5e+22)
		tmp = Float64(x - Float64(y * a));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.7e+41)
		tmp = x - a;
	elseif (z <= 7.5e+22)
		tmp = x - (y * a);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.7e+41], N[(x - a), $MachinePrecision], If[LessEqual[z, 7.5e+22], N[(x - N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.69999999999999981e41 or 7.5000000000000002e22 < z

   1. Initial program 95.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 73.6%

      \[\leadsto x - \color{blue}{a} \]

   if -3.69999999999999981e41 < z < 7.5000000000000002e22

   1. Initial program 99.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 72.9

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

   5. Simplified 72.9%

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

   6. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

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

      2. *-lowering-*.f64 67.3

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

   8. Simplified 67.3%

      \[\leadsto \color{blue}{x - a \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+41}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+22}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-77}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e-77) (- x a) (if (<= z 7.5e+22) x (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e-77) {
		tmp = x - a;
	} else if (z <= 7.5e+22) {
		tmp = x;
	} else {
		tmp = x - a;
	}
	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 (z <= (-5.5d-77)) then
        tmp = x - a
    else if (z <= 7.5d+22) then
        tmp = x
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e-77) {
		tmp = x - a;
	} else if (z <= 7.5e+22) {
		tmp = x;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e-77:
		tmp = x - a
	elif z <= 7.5e+22:
		tmp = x
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e-77)
		tmp = Float64(x - a);
	elseif (z <= 7.5e+22)
		tmp = x;
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.5e-77)
		tmp = x - a;
	elseif (z <= 7.5e+22)
		tmp = x;
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e-77], N[(x - a), $MachinePrecision], If[LessEqual[z, 7.5e+22], x, N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.49999999999999998e-77 or 7.5000000000000002e22 < z

   1. Initial program 95.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 69.5%

      \[\leadsto x - \color{blue}{a} \]

   if -5.49999999999999998e-77 < z < 7.5000000000000002e22

   1. Initial program 99.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 61.0%

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

Alternative 13: 53.9% accurate, 35.0× 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 97.7%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 53.1%

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

Developer Target 1: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\left(t - z\right) + 1} \cdot a \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (* (/ (- y z) (+ (- t z) 1.0)) a)))

C

double code(double x, double y, double z, double t, double a) {
	return x - (((y - z) / ((t - z) + 1.0)) * a);
}

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 - z) + 1.0d0)) * a)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - (((y - z) / ((t - z) + 1.0)) * a);
}

Python

def code(x, y, z, t, a):
	return x - (((y - z) / ((t - z) + 1.0)) * a)

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(Float64(y - z) / Float64(Float64(t - z) + 1.0)) * a))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - (((y - z) / ((t - z) + 1.0)) * a);
end

Wolfram

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (/ (- y z) (+ (- t z) 1)) a)))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))