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

Percentage Accurate: 97.1% → 97.3%

Time: 10.7s

Alternatives: 12

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: 97.1% 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: 97.3% 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 96.9%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
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 97.2%

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

5. Final simplification 97.2%

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

Alternative 2: 65.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-113}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_1 -2e-113)
     (- x a)
     (if (<= t_1 5e-36) x (if (<= t_1 5e+306) (- x a) (* a (- y)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -2e-113) {
		tmp = x - a;
	} else if (t_1 <= 5e-36) {
		tmp = x;
	} else if (t_1 <= 5e+306) {
		tmp = x - a;
	} else {
		tmp = a * -y;
	}
	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) :: tmp
    t_1 = (y - z) / (((t - z) + 1.0d0) / a)
    if (t_1 <= (-2d-113)) then
        tmp = x - a
    else if (t_1 <= 5d-36) then
        tmp = x
    else if (t_1 <= 5d+306) then
        tmp = x - a
    else
        tmp = a * -y
    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 - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -2e-113) {
		tmp = x - a;
	} else if (t_1 <= 5e-36) {
		tmp = x;
	} else if (t_1 <= 5e+306) {
		tmp = x - a;
	} else {
		tmp = a * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_1 <= -2e-113:
		tmp = x - a
	elif t_1 <= 5e-36:
		tmp = x
	elif t_1 <= 5e+306:
		tmp = x - a
	else:
		tmp = a * -y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_1 <= -2e-113)
		tmp = Float64(x - a);
	elseif (t_1 <= 5e-36)
		tmp = x;
	elseif (t_1 <= 5e+306)
		tmp = Float64(x - a);
	else
		tmp = Float64(a * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_1 <= -2e-113)
		tmp = x - a;
	elseif (t_1 <= 5e-36)
		tmp = x;
	elseif (t_1 <= 5e+306)
		tmp = x - a;
	else
		tmp = a * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-113], N[(x - a), $MachinePrecision], If[LessEqual[t$95$1, 5e-36], x, If[LessEqual[t$95$1, 5e+306], N[(x - a), $MachinePrecision], N[(a * (-y)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -1.99999999999999996e-113 or 5.00000000000000004e-36 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 4.99999999999999993e306

   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 57.9%

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

   if -1.99999999999999996e-113 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 5.00000000000000004e-36

   1. Initial program 93.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 86.0%

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

   if 4.99999999999999993e306 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 100.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. /-lowering-/.f64 N/A

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

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

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

      14. --lowering--.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-lowering-+.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

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

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

       3. mul-1-neg N/A

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

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

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

       5. mul-1-neg N/A

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

       6. neg-lowering-neg.f64 92.9

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

   11. Simplified 92.9%

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

Alternative 3: 56.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+51}:\\ \;\;\;\;-a\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_1 -4e+51) (- a) (if (<= t_1 2e+100) x (- a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -4e+51) {
		tmp = -a;
	} else if (t_1 <= 2e+100) {
		tmp = x;
	} else {
		tmp = -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) :: t_1
    real(8) :: tmp
    t_1 = (y - z) / (((t - z) + 1.0d0) / a)
    if (t_1 <= (-4d+51)) then
        tmp = -a
    else if (t_1 <= 2d+100) then
        tmp = x
    else
        tmp = -a
    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 - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -4e+51) {
		tmp = -a;
	} else if (t_1 <= 2e+100) {
		tmp = x;
	} else {
		tmp = -a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_1 <= -4e+51:
		tmp = -a
	elif t_1 <= 2e+100:
		tmp = x
	else:
		tmp = -a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_1 <= -4e+51)
		tmp = Float64(-a);
	elseif (t_1 <= 2e+100)
		tmp = x;
	else
		tmp = Float64(-a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_1 <= -4e+51)
		tmp = -a;
	elseif (t_1 <= 2e+100)
		tmp = x;
	else
		tmp = -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+51], (-a), If[LessEqual[t$95$1, 2e+100], x, (-a)]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -4e51 or 2.00000000000000003e100 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   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 42.8%

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

   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 38.3

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

   8. Simplified 38.3%

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

   if -4e51 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 2.00000000000000003e100

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.4%

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

Alternative 4: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-a}{t}, y - z, x\right)\\ \mathbf{if}\;t \leq -1.22 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{-1 + z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- a) t) (- y z) x)))
   (if (<= t -1.22e+143)
     t_1
     (if (<= t 6.2e+34) (fma (- y z) (/ a (+ -1.0 z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-a / t), (y - z), x);
	double tmp;
	if (t <= -1.22e+143) {
		tmp = t_1;
	} else if (t <= 6.2e+34) {
		tmp = fma((y - z), (a / (-1.0 + z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-a) / t), Float64(y - z), x)
	tmp = 0.0
	if (t <= -1.22e+143)
		tmp = t_1;
	elseif (t <= 6.2e+34)
		tmp = fma(Float64(y - z), Float64(a / Float64(-1.0 + z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-a) / t), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.22e+143], t$95$1, If[LessEqual[t, 6.2e+34], N[(N[(y - z), $MachinePrecision] * N[(a / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.22000000000000004e143 or 6.19999999999999955e34 < t

   1. Initial program 93.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 94.3%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 86.5

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

   7. Simplified 86.5%

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

   if -1.22000000000000004e143 < t < 6.19999999999999955e34

   1. Initial program 98.7%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 97.6

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

   5. Simplified 97.6%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{t}, y - z, x\right)\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{-1 + z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{t}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6e+75)
   (fma a (/ z (- t z)) x)
   (if (<= z 3.25e-54)
     (fma a (/ y (- -1.0 t)) x)
     (fma a (/ z (+ t (- 1.0 z))) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e+75) {
		tmp = fma(a, (z / (t - z)), x);
	} else if (z <= 3.25e-54) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = fma(a, (z / (t + (1.0 - z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6e+75)
		tmp = fma(a, Float64(z / Float64(t - z)), x);
	elseif (z <= 3.25e-54)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = fma(a, Float64(z / Float64(t + Float64(1.0 - z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6e+75], N[(a * N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3.25e-54], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(z / N[(t + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6e75

   1. Initial program 94.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 90.7

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

   5. Simplified 90.7%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 90.7

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

   8. Simplified 90.7%

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

   if -6e75 < z < 3.24999999999999996e-54

   1. Initial program 97.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 94.3

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

   5. Simplified 94.3%

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

   if 3.24999999999999996e-54 < z

   1. Initial program 97.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 86.9

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

   5. Simplified 86.9%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.12 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ z (- t z)) x)))
   (if (<= z -1.1e+76)
     t_1
     (if (<= z 2.12e-13) (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 / (t - z)), x);
	double tmp;
	if (z <= -1.1e+76) {
		tmp = t_1;
	} else if (z <= 2.12e-13) {
		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(a, Float64(z / Float64(t - z)), x)
	tmp = 0.0
	if (z <= -1.1e+76)
		tmp = t_1;
	elseif (z <= 2.12e-13)
		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[(a * N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.1e+76], t$95$1, If[LessEqual[z, 2.12e-13], 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 < -1.1e76 or 2.1200000000000001e-13 < z

   1. Initial program 96.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 89.1

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

   5. Simplified 89.1%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 89.1

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

   8. Simplified 89.1%

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

   if -1.1e76 < z < 2.1200000000000001e-13

   1. Initial program 97.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 93.2

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

   5. Simplified 93.2%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{elif}\;z \leq 2.12 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{t - z}, x\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, y - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e+76)
   (fma z (/ a (- t z)) x)
   (if (<= z 2.3e-13) (fma a (/ y (- -1.0 t)) x) (fma (/ a z) (- y z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e+76) {
		tmp = fma(z, (a / (t - z)), x);
	} else if (z <= 2.3e-13) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = fma((a / z), (y - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e+76)
		tmp = fma(z, Float64(a / Float64(t - z)), x);
	elseif (z <= 2.3e-13)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = fma(Float64(a / z), Float64(y - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e+76], N[(z * N[(a / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.3e-13], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(a / z), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6999999999999999e76

   1. Initial program 94.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 90.7

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

   5. Simplified 90.7%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 90.7

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

   8. Simplified 90.7%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

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

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

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

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

       6. --lowering--.f64 86.6

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

   11. Simplified 86.6%

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

   if -2.6999999999999999e76 < z < 2.29999999999999979e-13

   1. Initial program 97.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 93.2

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

   5. Simplified 93.2%

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

   if 2.29999999999999979e-13 < z

   1. Initial program 97.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 97.2%

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

   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 87.6

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

   7. Simplified 87.6%

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

Alternative 8: 86.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{a}{t - z}, x\right)\\ \mathbf{if}\;z \leq -1.28 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ a (- t z)) x)))
   (if (<= z -1.28e+76)
     t_1
     (if (<= z 2.3e-13) (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(z, (a / (t - z)), x);
	double tmp;
	if (z <= -1.28e+76) {
		tmp = t_1;
	} else if (z <= 2.3e-13) {
		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(z, Float64(a / Float64(t - z)), x)
	tmp = 0.0
	if (z <= -1.28e+76)
		tmp = t_1;
	elseif (z <= 2.3e-13)
		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[(z * N[(a / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.28e+76], t$95$1, If[LessEqual[z, 2.3e-13], 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 < -1.27999999999999994e76 or 2.29999999999999979e-13 < z

   1. Initial program 96.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 89.1

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

   5. Simplified 89.1%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 89.1

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

   8. Simplified 89.1%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

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

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

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

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

       6. --lowering--.f64 85.8

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

   11. Simplified 85.8%

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

   if -1.27999999999999994e76 < z < 2.29999999999999979e-13

   1. Initial program 97.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 93.2

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

   5. Simplified 93.2%

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

Alternative 9: 84.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+81}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.5e+81)
   (- x a)
   (if (<= z 2.3e-13) (fma a (/ y (- -1.0 t)) x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e+81) {
		tmp = x - a;
	} else if (z <= 2.3e-13) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.5e+81)
		tmp = Float64(x - a);
	elseif (z <= 2.3e-13)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.5e+81], N[(x - a), $MachinePrecision], If[LessEqual[z, 2.3e-13], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.49999999999999986e81 or 2.29999999999999979e-13 < z

   1. Initial program 96.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 81.1%

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

   if -8.49999999999999986e81 < z < 2.29999999999999979e-13

   1. Initial program 97.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 92.5

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

   5. Simplified 92.5%

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

Alternative 10: 73.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+75}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-13}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.3e+75) (- x a) (if (<= z 2.3e-13) (- x (* a y)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.3e+75) {
		tmp = x - a;
	} else if (z <= 2.3e-13) {
		tmp = x - (a * y);
	} 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 <= (-5.3d+75)) then
        tmp = x - a
    else if (z <= 2.3d-13) then
        tmp = x - (a * y)
    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 <= -5.3e+75) {
		tmp = x - a;
	} else if (z <= 2.3e-13) {
		tmp = x - (a * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.3e+75:
		tmp = x - a
	elif z <= 2.3e-13:
		tmp = x - (a * y)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.3e+75)
		tmp = Float64(x - a);
	elseif (z <= 2.3e-13)
		tmp = Float64(x - Float64(a * y));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.3e+75)
		tmp = x - a;
	elseif (z <= 2.3e-13)
		tmp = x - (a * y);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.3e+75], N[(x - a), $MachinePrecision], If[LessEqual[z, 2.3e-13], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2999999999999998e75 or 2.29999999999999979e-13 < z

   1. Initial program 96.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 80.5%

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

   if -5.2999999999999998e75 < z < 2.29999999999999979e-13

   1. Initial program 97.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 77.5

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

   5. Simplified 77.5%

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

   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 75.7

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

   8. Simplified 75.7%

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

Alternative 11: 66.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.25 \cdot 10^{-10}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 250000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.25e-10) (- x a) (if (<= z 250000000000.0) x (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.25e-10) {
		tmp = x - a;
	} else if (z <= 250000000000.0) {
		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 <= (-4.25d-10)) then
        tmp = x - a
    else if (z <= 250000000000.0d0) 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 <= -4.25e-10) {
		tmp = x - a;
	} else if (z <= 250000000000.0) {
		tmp = x;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.25e-10:
		tmp = x - a
	elif z <= 250000000000.0:
		tmp = x
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.25e-10)
		tmp = Float64(x - a);
	elseif (z <= 250000000000.0)
		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 <= -4.25e-10)
		tmp = x - a;
	elseif (z <= 250000000000.0)
		tmp = x;
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.25e-10], N[(x - a), $MachinePrecision], If[LessEqual[z, 250000000000.0], x, N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2499999999999998e-10 or 2.5e11 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 78.7%

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

   if -4.2499999999999998e-10 < z < 2.5e11

   1. Initial program 98.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 57.9%

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

Alternative 12: 52.9% 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 96.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 49.9%

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

Developer Target 1: 99.6% 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 2024204 
(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))))