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

Percentage Accurate: 97.2% → 99.6%

Time: 12.9s

Alternatives: 11

Speedup: 1.0×

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: 97.2% 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: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. sub-neg 98.3%

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

   2. +-commutative 98.3%

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

   3. associate-/r/ 99.9%

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

   4. distribute-lft-neg-in 99.9%

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

   5. fma-def 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. sub-neg 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. remove-double-neg 99.9%

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

   10. +-commutative 99.9%

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

   11. sub-neg 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot a\\ t_2 := x - \frac{a}{\frac{t}{y}}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+72}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{-148}:\\ \;\;\;\;x + z \cdot a\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-235}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 0.000102:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y a))) (t_2 (- x (/ a (/ t y)))))
   (if (<= z -3.4e+72)
     (- x a)
     (if (<= z -5.6e-46)
       t_2
       (if (<= z -2.75e-148)
         (+ x (* z a))
         (if (<= z -1.8e-202)
           t_1
           (if (<= z -9.8e-235) t_2 (if (<= z 0.000102) t_1 (- x a)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * a);
	double t_2 = x - (a / (t / y));
	double tmp;
	if (z <= -3.4e+72) {
		tmp = x - a;
	} else if (z <= -5.6e-46) {
		tmp = t_2;
	} else if (z <= -2.75e-148) {
		tmp = x + (z * a);
	} else if (z <= -1.8e-202) {
		tmp = t_1;
	} else if (z <= -9.8e-235) {
		tmp = t_2;
	} else if (z <= 0.000102) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x - (y * a)
    t_2 = x - (a / (t / y))
    if (z <= (-3.4d+72)) then
        tmp = x - a
    else if (z <= (-5.6d-46)) then
        tmp = t_2
    else if (z <= (-2.75d-148)) then
        tmp = x + (z * a)
    else if (z <= (-1.8d-202)) then
        tmp = t_1
    else if (z <= (-9.8d-235)) then
        tmp = t_2
    else if (z <= 0.000102d0) then
        tmp = t_1
    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 t_1 = x - (y * a);
	double t_2 = x - (a / (t / y));
	double tmp;
	if (z <= -3.4e+72) {
		tmp = x - a;
	} else if (z <= -5.6e-46) {
		tmp = t_2;
	} else if (z <= -2.75e-148) {
		tmp = x + (z * a);
	} else if (z <= -1.8e-202) {
		tmp = t_1;
	} else if (z <= -9.8e-235) {
		tmp = t_2;
	} else if (z <= 0.000102) {
		tmp = t_1;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - (y * a)
	t_2 = x - (a / (t / y))
	tmp = 0
	if z <= -3.4e+72:
		tmp = x - a
	elif z <= -5.6e-46:
		tmp = t_2
	elif z <= -2.75e-148:
		tmp = x + (z * a)
	elif z <= -1.8e-202:
		tmp = t_1
	elif z <= -9.8e-235:
		tmp = t_2
	elif z <= 0.000102:
		tmp = t_1
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * a))
	t_2 = Float64(x - Float64(a / Float64(t / y)))
	tmp = 0.0
	if (z <= -3.4e+72)
		tmp = Float64(x - a);
	elseif (z <= -5.6e-46)
		tmp = t_2;
	elseif (z <= -2.75e-148)
		tmp = Float64(x + Float64(z * a));
	elseif (z <= -1.8e-202)
		tmp = t_1;
	elseif (z <= -9.8e-235)
		tmp = t_2;
	elseif (z <= 0.000102)
		tmp = t_1;
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - (y * a);
	t_2 = x - (a / (t / y));
	tmp = 0.0;
	if (z <= -3.4e+72)
		tmp = x - a;
	elseif (z <= -5.6e-46)
		tmp = t_2;
	elseif (z <= -2.75e-148)
		tmp = x + (z * a);
	elseif (z <= -1.8e-202)
		tmp = t_1;
	elseif (z <= -9.8e-235)
		tmp = t_2;
	elseif (z <= 0.000102)
		tmp = t_1;
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(a / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.4e+72], N[(x - a), $MachinePrecision], If[LessEqual[z, -5.6e-46], t$95$2, If[LessEqual[z, -2.75e-148], N[(x + N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.8e-202], t$95$1, If[LessEqual[z, -9.8e-235], t$95$2, If[LessEqual[z, 0.000102], t$95$1, N[(x - a), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.3999999999999998e72 or 1.01999999999999999e-4 < z

   1. Initial program 96.0%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 82.3%

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

   if -3.3999999999999998e72 < z < -5.5999999999999997e-46 or -1.8000000000000001e-202 < z < -9.79999999999999931e-235

   1. Initial program 99.8%

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

      1. associate-/r/ 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around inf 72.4%

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

      1. associate-/l* 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in y around inf 82.1%

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

   if -5.5999999999999997e-46 < z < -2.7500000000000001e-148

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-/r/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. sub-neg 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 84.4%

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

   5. Taylor expanded in y around 0 74.8%

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

   6. Taylor expanded in z around 0 74.7%

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

   if -2.7500000000000001e-148 < z < -1.8000000000000001e-202 or -9.79999999999999931e-235 < z < 1.01999999999999999e-4

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. sub-neg 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. remove-double-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 82.2%

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

   5. Taylor expanded in z around 0 78.7%

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

      1. mul-1-neg 78.7%

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

      2. unsub-neg 78.7%

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

   7. Simplified 78.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+72}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-46}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{-148}:\\ \;\;\;\;x + z \cdot a\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-202}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-235}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 0.000102:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 3: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y - z}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+27}:\\ \;\;\;\;x - \frac{a}{\frac{1 - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -6e+18)
   (- x (/ a (/ t (- y z))))
   (if (<= t 3.7e+27) (- x (/ a (/ (- 1.0 z) (- y z)))) (- x (/ a (/ t y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -6e+18) {
		tmp = x - (a / (t / (y - z)));
	} else if (t <= 3.7e+27) {
		tmp = x - (a / ((1.0 - z) / (y - z)));
	} else {
		tmp = x - (a / (t / 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) :: tmp
    if (t <= (-6d+18)) then
        tmp = x - (a / (t / (y - z)))
    else if (t <= 3.7d+27) then
        tmp = x - (a / ((1.0d0 - z) / (y - z)))
    else
        tmp = x - (a / (t / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -6e+18) {
		tmp = x - (a / (t / (y - z)));
	} else if (t <= 3.7e+27) {
		tmp = x - (a / ((1.0 - z) / (y - z)));
	} else {
		tmp = x - (a / (t / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -6e+18:
		tmp = x - (a / (t / (y - z)))
	elif t <= 3.7e+27:
		tmp = x - (a / ((1.0 - z) / (y - z)))
	else:
		tmp = x - (a / (t / y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -6e+18)
		tmp = Float64(x - Float64(a / Float64(t / Float64(y - z))));
	elseif (t <= 3.7e+27)
		tmp = Float64(x - Float64(a / Float64(Float64(1.0 - z) / Float64(y - z))));
	else
		tmp = Float64(x - Float64(a / Float64(t / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -6e+18)
		tmp = x - (a / (t / (y - z)));
	elseif (t <= 3.7e+27)
		tmp = x - (a / ((1.0 - z) / (y - z)));
	else
		tmp = x - (a / (t / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -6e+18], N[(x - N[(a / N[(t / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.7e+27], N[(x - N[(a / N[(N[(1.0 - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(a / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6e18

   1. Initial program 99.9%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 80.1%

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

      1. associate-/l* 92.4%

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

   6. Simplified 92.4%

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

   if -6e18 < t < 3.70000000000000002e27

   1. Initial program 98.3%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 87.9%

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

      1. associate-/l* 98.6%

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

   6. Simplified 98.6%

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

   if 3.70000000000000002e27 < t

   1. Initial program 96.5%

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

      1. associate-/r/ 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around inf 74.6%

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

      1. associate-/l* 88.0%

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

   6. Simplified 88.0%

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

   7. Taylor expanded in y around inf 88.1%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y - z}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+27}:\\ \;\;\;\;x - \frac{a}{\frac{1 - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \end{array} \]

Alternative 4: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+73}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 0.000102:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.2e+73)
   (- x a)
   (if (<= z 0.000102) (- x (* a (/ y (+ t 1.0)))) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e+73) {
		tmp = x - a;
	} else if (z <= 0.000102) {
		tmp = x - (a * (y / (t + 1.0)));
	} 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 <= (-5.2d+73)) then
        tmp = x - a
    else if (z <= 0.000102d0) then
        tmp = x - (a * (y / (t + 1.0d0)))
    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 <= -5.2e+73) {
		tmp = x - a;
	} else if (z <= 0.000102) {
		tmp = x - (a * (y / (t + 1.0)));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.2e+73:
		tmp = x - a
	elif z <= 0.000102:
		tmp = x - (a * (y / (t + 1.0)))
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2e+73)
		tmp = Float64(x - a);
	elseif (z <= 0.000102)
		tmp = Float64(x - Float64(a * Float64(y / Float64(t + 1.0))));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.2e+73)
		tmp = x - a;
	elseif (z <= 0.000102)
		tmp = x - (a * (y / (t + 1.0)));
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2e+73], N[(x - a), $MachinePrecision], If[LessEqual[z, 0.000102], N[(x - N[(a * N[(y / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2000000000000001e73 or 1.01999999999999999e-4 < z

   1. Initial program 96.0%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 82.3%

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

   if -5.2000000000000001e73 < z < 1.01999999999999999e-4

   1. Initial program 99.8%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 91.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+73}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 0.000102:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 5: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -26000000000:\\ \;\;\;\;x + \frac{z - y}{\frac{-z}{a}}\\ \mathbf{elif}\;z \leq 0.000102:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -26000000000.0)
   (+ x (/ (- z y) (/ (- z) a)))
   (if (<= z 0.000102) (- x (* a (/ y (+ t 1.0)))) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -26000000000.0) {
		tmp = x + ((z - y) / (-z / a));
	} else if (z <= 0.000102) {
		tmp = x - (a * (y / (t + 1.0)));
	} 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 <= (-26000000000.0d0)) then
        tmp = x + ((z - y) / (-z / a))
    else if (z <= 0.000102d0) then
        tmp = x - (a * (y / (t + 1.0d0)))
    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 <= -26000000000.0) {
		tmp = x + ((z - y) / (-z / a));
	} else if (z <= 0.000102) {
		tmp = x - (a * (y / (t + 1.0)));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -26000000000.0:
		tmp = x + ((z - y) / (-z / a))
	elif z <= 0.000102:
		tmp = x - (a * (y / (t + 1.0)))
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -26000000000.0)
		tmp = Float64(x + Float64(Float64(z - y) / Float64(Float64(-z) / a)));
	elseif (z <= 0.000102)
		tmp = Float64(x - Float64(a * Float64(y / Float64(t + 1.0))));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -26000000000.0)
		tmp = x + ((z - y) / (-z / a));
	elseif (z <= 0.000102)
		tmp = x - (a * (y / (t + 1.0)));
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -26000000000.0], N[(x + N[(N[(z - y), $MachinePrecision] / N[((-z) / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.000102], N[(x - N[(a * N[(y / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6e10

   1. Initial program 98.0%

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

   2. Taylor expanded in z around inf 88.9%

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

      1. mul-1-neg 88.9%

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

      2. distribute-neg-frac 88.9%

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

   4. Simplified 88.9%

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

   if -2.6e10 < z < 1.01999999999999999e-4

   1. Initial program 99.8%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 93.1%

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

   if 1.01999999999999999e-4 < z

   1. Initial program 94.2%

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

      1. associate-/r/ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 83.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -26000000000:\\ \;\;\;\;x + \frac{z - y}{\frac{-z}{a}}\\ \mathbf{elif}\;z \leq 0.000102:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (a * ((z - y) / ((t - z) + 1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. associate-/r/ 99.9%

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

   2. *-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.25 \cdot 10^{-9}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;x + z \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.25e-9) (- x a) (if (<= z 1.9e-52) (+ x (* z a)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.25e-9) {
		tmp = x - a;
	} else if (z <= 1.9e-52) {
		tmp = x + (z * 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 <= (-3.25d-9)) then
        tmp = x - a
    else if (z <= 1.9d-52) then
        tmp = x + (z * 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 <= -3.25e-9) {
		tmp = x - a;
	} else if (z <= 1.9e-52) {
		tmp = x + (z * a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.25e-9:
		tmp = x - a
	elif z <= 1.9e-52:
		tmp = x + (z * a)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.25e-9)
		tmp = Float64(x - a);
	elseif (z <= 1.9e-52)
		tmp = Float64(x + Float64(z * 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 <= -3.25e-9)
		tmp = x - a;
	elseif (z <= 1.9e-52)
		tmp = x + (z * a);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.25e-9], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.9e-52], N[(x + N[(z * a), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2500000000000002e-9 or 1.9000000000000002e-52 < z

   1. Initial program 96.7%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 78.2%

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

   if -3.2500000000000002e-9 < z < 1.9000000000000002e-52

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. sub-neg 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. remove-double-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 77.0%

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

   5. Taylor expanded in y around 0 59.9%

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

   6. Taylor expanded in z around 0 59.9%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.25 \cdot 10^{-9}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;x + z \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 8: 72.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+50}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 0.0001:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.6e+50) (- x a) (if (<= z 0.0001) (- x (* y a)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e+50) {
		tmp = x - a;
	} else if (z <= 0.0001) {
		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 <= (-3.6d+50)) then
        tmp = x - a
    else if (z <= 0.0001d0) 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 <= -3.6e+50) {
		tmp = x - a;
	} else if (z <= 0.0001) {
		tmp = x - (y * a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.6e+50:
		tmp = x - a
	elif z <= 0.0001:
		tmp = x - (y * a)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.6e+50)
		tmp = Float64(x - a);
	elseif (z <= 0.0001)
		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 <= -3.6e+50)
		tmp = x - a;
	elseif (z <= 0.0001)
		tmp = x - (y * a);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.6e+50], N[(x - a), $MachinePrecision], If[LessEqual[z, 0.0001], N[(x - N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999986e50 or 1.00000000000000005e-4 < z

   1. Initial program 96.1%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 81.8%

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

   if -3.59999999999999986e50 < z < 1.00000000000000005e-4

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. sub-neg 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. remove-double-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 78.0%

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

   5. Taylor expanded in z around 0 72.4%

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

      1. mul-1-neg 72.4%

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

      2. unsub-neg 72.4%

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

   7. Simplified 72.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+50}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 0.0001:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 9: 65.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-9}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 620000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.8e-9) (- x a) (if (<= z 620000000.0) x (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.8e-9) {
		tmp = x - a;
	} else if (z <= 620000000.0) {
		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 <= (-1.8d-9)) then
        tmp = x - a
    else if (z <= 620000000.0d0) 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 <= -1.8e-9) {
		tmp = x - a;
	} else if (z <= 620000000.0) {
		tmp = x;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.8e-9:
		tmp = x - a
	elif z <= 620000000.0:
		tmp = x
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.8e-9)
		tmp = Float64(x - a);
	elseif (z <= 620000000.0)
		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 <= -1.8e-9)
		tmp = x - a;
	elseif (z <= 620000000.0)
		tmp = x;
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.8e-9], N[(x - a), $MachinePrecision], If[LessEqual[z, 620000000.0], x, N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e-9 or 6.2e8 < z

   1. Initial program 96.4%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 78.9%

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

   if -1.8e-9 < z < 6.2e8

   1. Initial program 99.8%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 60.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-9}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 620000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 10: 53.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+135}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= a -1.2e+135) (- a) x))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.2e+135) {
		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 (a <= (-1.2d+135)) 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 (a <= -1.2e+135) {
		tmp = -a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.2e+135:
		tmp = -a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.2e+135)
		tmp = Float64(-a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.2e+135)
		tmp = -a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.2e+135], (-a), x]

Derivation

1. Split input into 2 regimes

2. if a < -1.19999999999999999e135

   1. Initial program 99.8%

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

      1. associate-/r/ 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around inf 89.7%

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

      1. associate--l+ 89.7%

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

      2. +-commutative 89.7%

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

      3. associate--l+ 89.7%

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

      4. +-commutative 89.7%

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

      5. div-sub 89.7%

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

      6. +-commutative 89.7%

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

   6. Simplified 89.7%

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

   7. Taylor expanded in z around inf 36.4%

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

      1. neg-mul-1 36.4%

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

   9. Simplified 36.4%

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

   if -1.19999999999999999e135 < a

   1. Initial program 98.0%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 67.1%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+135}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 52.6% accurate, 13.0× 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 98.3%

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

   1. associate-/r/ 99.9%

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

   2. *-commutative 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 58.3%

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

5. Final simplification 58.3%

   \[\leadsto x \]

Developer target: 99.6% accurate, 1.0× 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 2023279 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1.0)) a))

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