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

Percentage Accurate: 97.0% → 99.7%

Time: 10.8s

Alternatives: 10

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: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× 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 96.1%

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

   1. associate-/r/ 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot a\\ t_2 := x - \frac{a}{t} \cdot \left(y - z\right)\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-94}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq -2.35 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.0009:\\ \;\;\;\;x + \frac{z \cdot a}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y a))) (t_2 (- x (* (/ a t) (- y z)))))
   (if (<= t -1.85e+18)
     t_2
     (if (<= t -1.8e-94)
       (- x a)
       (if (<= t -2.35e-142)
         t_1
         (if (<= t 7.6e-150)
           (- x a)
           (if (<= t 4e-70)
             t_1
             (if (<= t 0.0009) (+ x (/ (* z a) (- 1.0 z))) t_2))))))))

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 - z));
	double tmp;
	if (t <= -1.85e+18) {
		tmp = t_2;
	} else if (t <= -1.8e-94) {
		tmp = x - a;
	} else if (t <= -2.35e-142) {
		tmp = t_1;
	} else if (t <= 7.6e-150) {
		tmp = x - a;
	} else if (t <= 4e-70) {
		tmp = t_1;
	} else if (t <= 0.0009) {
		tmp = x + ((z * a) / (1.0 - z));
	} else {
		tmp = t_2;
	}
	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 - z))
    if (t <= (-1.85d+18)) then
        tmp = t_2
    else if (t <= (-1.8d-94)) then
        tmp = x - a
    else if (t <= (-2.35d-142)) then
        tmp = t_1
    else if (t <= 7.6d-150) then
        tmp = x - a
    else if (t <= 4d-70) then
        tmp = t_1
    else if (t <= 0.0009d0) then
        tmp = x + ((z * a) / (1.0d0 - z))
    else
        tmp = t_2
    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 - z));
	double tmp;
	if (t <= -1.85e+18) {
		tmp = t_2;
	} else if (t <= -1.8e-94) {
		tmp = x - a;
	} else if (t <= -2.35e-142) {
		tmp = t_1;
	} else if (t <= 7.6e-150) {
		tmp = x - a;
	} else if (t <= 4e-70) {
		tmp = t_1;
	} else if (t <= 0.0009) {
		tmp = x + ((z * a) / (1.0 - z));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - (y * a)
	t_2 = x - ((a / t) * (y - z))
	tmp = 0
	if t <= -1.85e+18:
		tmp = t_2
	elif t <= -1.8e-94:
		tmp = x - a
	elif t <= -2.35e-142:
		tmp = t_1
	elif t <= 7.6e-150:
		tmp = x - a
	elif t <= 4e-70:
		tmp = t_1
	elif t <= 0.0009:
		tmp = x + ((z * a) / (1.0 - z))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * a))
	t_2 = Float64(x - Float64(Float64(a / t) * Float64(y - z)))
	tmp = 0.0
	if (t <= -1.85e+18)
		tmp = t_2;
	elseif (t <= -1.8e-94)
		tmp = Float64(x - a);
	elseif (t <= -2.35e-142)
		tmp = t_1;
	elseif (t <= 7.6e-150)
		tmp = Float64(x - a);
	elseif (t <= 4e-70)
		tmp = t_1;
	elseif (t <= 0.0009)
		tmp = Float64(x + Float64(Float64(z * a) / Float64(1.0 - z)));
	else
		tmp = t_2;
	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 - z));
	tmp = 0.0;
	if (t <= -1.85e+18)
		tmp = t_2;
	elseif (t <= -1.8e-94)
		tmp = x - a;
	elseif (t <= -2.35e-142)
		tmp = t_1;
	elseif (t <= 7.6e-150)
		tmp = x - a;
	elseif (t <= 4e-70)
		tmp = t_1;
	elseif (t <= 0.0009)
		tmp = x + ((z * a) / (1.0 - z));
	else
		tmp = t_2;
	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[(N[(a / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.85e+18], t$95$2, If[LessEqual[t, -1.8e-94], N[(x - a), $MachinePrecision], If[LessEqual[t, -2.35e-142], t$95$1, If[LessEqual[t, 7.6e-150], N[(x - a), $MachinePrecision], If[LessEqual[t, 4e-70], t$95$1, If[LessEqual[t, 0.0009], N[(x + N[(N[(z * a), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.85e18 or 8.9999999999999998e-4 < t

   1. Initial program 95.5%

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

      1. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 82.0%

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

      1. associate-/l* 89.3%

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

      2. associate-/r/ 88.1%

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

   7. Simplified 88.1%

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

   if -1.85e18 < t < -1.8e-94 or -2.34999999999999995e-142 < t < 7.5999999999999997e-150

   1. Initial program 97.7%

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

      1. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 71.5%

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

   if -1.8e-94 < t < -2.34999999999999995e-142 or 7.5999999999999997e-150 < t < 3.99999999999999998e-70

   1. Initial program 94.5%

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

      1. associate-/r/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 99.8%

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

   6. Taylor expanded in z around 0 73.6%

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

   if 3.99999999999999998e-70 < t < 8.9999999999999998e-4

   1. Initial program 94.6%

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

      1. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 96.9%

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

   6. Taylor expanded in y around 0 81.4%

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

      1. cancel-sign-sub-inv 81.4%

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

      2. metadata-eval 81.4%

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

      3. *-lft-identity 81.4%

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

   8. Simplified 81.4%

      \[\leadsto \color{blue}{x + \frac{a \cdot z}{1 - z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{a}{t} \cdot \left(y - z\right)\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-94}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq -2.35 \cdot 10^{-142}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-150}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-70}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;t \leq 0.0009:\\ \;\;\;\;x + \frac{z \cdot a}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{t} \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot a\\ t_2 := x - a \cdot \frac{y}{t}\\ \mathbf{if}\;t \leq -2.4:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-94}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-148}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+33}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y a))) (t_2 (- x (* a (/ y t)))))
   (if (<= t -2.4)
     t_2
     (if (<= t -3.6e-94)
       (- x a)
       (if (<= t -2e-143)
         t_1
         (if (<= t 5.2e-148)
           (- x a)
           (if (<= t 3.2e-64) t_1 (if (<= t 8.5e+33) (- x a) t_2))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * a);
	double t_2 = x - (a * (y / t));
	double tmp;
	if (t <= -2.4) {
		tmp = t_2;
	} else if (t <= -3.6e-94) {
		tmp = x - a;
	} else if (t <= -2e-143) {
		tmp = t_1;
	} else if (t <= 5.2e-148) {
		tmp = x - a;
	} else if (t <= 3.2e-64) {
		tmp = t_1;
	} else if (t <= 8.5e+33) {
		tmp = x - a;
	} else {
		tmp = t_2;
	}
	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 * (y / t))
    if (t <= (-2.4d0)) then
        tmp = t_2
    else if (t <= (-3.6d-94)) then
        tmp = x - a
    else if (t <= (-2d-143)) then
        tmp = t_1
    else if (t <= 5.2d-148) then
        tmp = x - a
    else if (t <= 3.2d-64) then
        tmp = t_1
    else if (t <= 8.5d+33) then
        tmp = x - a
    else
        tmp = t_2
    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 * (y / t));
	double tmp;
	if (t <= -2.4) {
		tmp = t_2;
	} else if (t <= -3.6e-94) {
		tmp = x - a;
	} else if (t <= -2e-143) {
		tmp = t_1;
	} else if (t <= 5.2e-148) {
		tmp = x - a;
	} else if (t <= 3.2e-64) {
		tmp = t_1;
	} else if (t <= 8.5e+33) {
		tmp = x - a;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - (y * a)
	t_2 = x - (a * (y / t))
	tmp = 0
	if t <= -2.4:
		tmp = t_2
	elif t <= -3.6e-94:
		tmp = x - a
	elif t <= -2e-143:
		tmp = t_1
	elif t <= 5.2e-148:
		tmp = x - a
	elif t <= 3.2e-64:
		tmp = t_1
	elif t <= 8.5e+33:
		tmp = x - a
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * a))
	t_2 = Float64(x - Float64(a * Float64(y / t)))
	tmp = 0.0
	if (t <= -2.4)
		tmp = t_2;
	elseif (t <= -3.6e-94)
		tmp = Float64(x - a);
	elseif (t <= -2e-143)
		tmp = t_1;
	elseif (t <= 5.2e-148)
		tmp = Float64(x - a);
	elseif (t <= 3.2e-64)
		tmp = t_1;
	elseif (t <= 8.5e+33)
		tmp = Float64(x - a);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - (y * a);
	t_2 = x - (a * (y / t));
	tmp = 0.0;
	if (t <= -2.4)
		tmp = t_2;
	elseif (t <= -3.6e-94)
		tmp = x - a;
	elseif (t <= -2e-143)
		tmp = t_1;
	elseif (t <= 5.2e-148)
		tmp = x - a;
	elseif (t <= 3.2e-64)
		tmp = t_1;
	elseif (t <= 8.5e+33)
		tmp = x - a;
	else
		tmp = t_2;
	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[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.4], t$95$2, If[LessEqual[t, -3.6e-94], N[(x - a), $MachinePrecision], If[LessEqual[t, -2e-143], t$95$1, If[LessEqual[t, 5.2e-148], N[(x - a), $MachinePrecision], If[LessEqual[t, 3.2e-64], t$95$1, If[LessEqual[t, 8.5e+33], N[(x - a), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.39999999999999991 or 8.4999999999999998e33 < t

   1. Initial program 95.3%

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

      1. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 88.4%

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

   6. Taylor expanded in t around inf 87.8%

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

   if -2.39999999999999991 < t < -3.6e-94 or -1.9999999999999999e-143 < t < 5.20000000000000015e-148 or 3.19999999999999975e-64 < t < 8.4999999999999998e33

   1. Initial program 97.2%

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

      1. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 72.0%

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

   if -3.6e-94 < t < -1.9999999999999999e-143 or 5.20000000000000015e-148 < t < 3.19999999999999975e-64

   1. Initial program 95.0%

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

      1. associate-/r/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 99.8%

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

   6. Taylor expanded in z around 0 71.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-94}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-143}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-148}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-64}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+33}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -510000 \lor \neg \left(z \leq 1.06 \cdot 10^{-7}\right):\\ \;\;\;\;x + a \cdot \frac{z - y}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - y}{\frac{t + 1}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -510000.0) (not (<= z 1.06e-7)))
   (+ x (* a (/ (- z y) (- 1.0 z))))
   (+ x (/ (- z y) (/ (+ t 1.0) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -510000.0) || !(z <= 1.06e-7)) {
		tmp = x + (a * ((z - y) / (1.0 - z)));
	} else {
		tmp = x + ((z - y) / ((t + 1.0) / 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 <= (-510000.0d0)) .or. (.not. (z <= 1.06d-7))) then
        tmp = x + (a * ((z - y) / (1.0d0 - z)))
    else
        tmp = x + ((z - y) / ((t + 1.0d0) / 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 <= -510000.0) || !(z <= 1.06e-7)) {
		tmp = x + (a * ((z - y) / (1.0 - z)));
	} else {
		tmp = x + ((z - y) / ((t + 1.0) / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -510000.0) or not (z <= 1.06e-7):
		tmp = x + (a * ((z - y) / (1.0 - z)))
	else:
		tmp = x + ((z - y) / ((t + 1.0) / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -510000.0) || !(z <= 1.06e-7))
		tmp = Float64(x + Float64(a * Float64(Float64(z - y) / Float64(1.0 - z))));
	else
		tmp = Float64(x + Float64(Float64(z - y) / Float64(Float64(t + 1.0) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -510000.0) || ~((z <= 1.06e-7)))
		tmp = x + (a * ((z - y) / (1.0 - z)));
	else
		tmp = x + ((z - y) / ((t + 1.0) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -510000.0], N[Not[LessEqual[z, 1.06e-7]], $MachinePrecision]], N[(x + N[(a * N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z - y), $MachinePrecision] / N[(N[(t + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1e5 or 1.06e-7 < z

   1. Initial program 92.8%

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

      1. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 91.0%

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

   if -5.1e5 < z < 1.06e-7

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0 99.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -510000 \lor \neg \left(z \leq 1.06 \cdot 10^{-7}\right):\\ \;\;\;\;x + a \cdot \frac{z - y}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - y}{\frac{t + 1}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+19}:\\ \;\;\;\;x - a \cdot \frac{y - z}{t}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;x + a \cdot \frac{z - y}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{t} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.1e+19)
   (- x (* a (/ (- y z) t)))
   (if (<= t 2.1e+31)
     (+ x (* a (/ (- z y) (- 1.0 z))))
     (- x (* (/ a t) (- y z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.1e+19) {
		tmp = x - (a * ((y - z) / t));
	} else if (t <= 2.1e+31) {
		tmp = x + (a * ((z - y) / (1.0 - z)));
	} else {
		tmp = x - ((a / t) * (y - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-1.1d+19)) then
        tmp = x - (a * ((y - z) / t))
    else if (t <= 2.1d+31) then
        tmp = x + (a * ((z - y) / (1.0d0 - z)))
    else
        tmp = x - ((a / t) * (y - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.1e+19) {
		tmp = x - (a * ((y - z) / t));
	} else if (t <= 2.1e+31) {
		tmp = x + (a * ((z - y) / (1.0 - z)));
	} else {
		tmp = x - ((a / t) * (y - z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.1e+19:
		tmp = x - (a * ((y - z) / t))
	elif t <= 2.1e+31:
		tmp = x + (a * ((z - y) / (1.0 - z)))
	else:
		tmp = x - ((a / t) * (y - z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.1e+19)
		tmp = Float64(x - Float64(a * Float64(Float64(y - z) / t)));
	elseif (t <= 2.1e+31)
		tmp = Float64(x + Float64(a * Float64(Float64(z - y) / Float64(1.0 - z))));
	else
		tmp = Float64(x - Float64(Float64(a / t) * Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.1e+19)
		tmp = x - (a * ((y - z) / t));
	elseif (t <= 2.1e+31)
		tmp = x + (a * ((z - y) / (1.0 - z)));
	else
		tmp = x - ((a / t) * (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.1e+19], N[(x - N[(a * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e+31], N[(x + N[(a * N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(a / t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1e19

   1. Initial program 92.2%

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

      1. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 91.2%

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

   if -1.1e19 < t < 2.09999999999999979e31

   1. Initial program 96.6%

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

      1. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 97.5%

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

   if 2.09999999999999979e31 < t

   1. Initial program 98.2%

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

      1. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 84.4%

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

      1. associate-/l* 91.1%

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

      2. associate-/r/ 91.1%

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

   7. Simplified 91.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+19}:\\ \;\;\;\;x - a \cdot \frac{y - z}{t}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;x + a \cdot \frac{z - y}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{t} \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -130000000 \lor \neg \left(z \leq 150\right):\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -130000000.0) (not (<= z 150.0)))
   (- (- x a) (/ a z))
   (- x (* a (/ y (+ t 1.0))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -130000000.0) || !(z <= 150.0)) {
		tmp = (x - a) - (a / z);
	} else {
		tmp = x - (a * (y / (t + 1.0)));
	}
	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 <= (-130000000.0d0)) .or. (.not. (z <= 150.0d0))) then
        tmp = (x - a) - (a / z)
    else
        tmp = x - (a * (y / (t + 1.0d0)))
    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 <= -130000000.0) || !(z <= 150.0)) {
		tmp = (x - a) - (a / z);
	} else {
		tmp = x - (a * (y / (t + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -130000000.0) or not (z <= 150.0):
		tmp = (x - a) - (a / z)
	else:
		tmp = x - (a * (y / (t + 1.0)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -130000000.0) || !(z <= 150.0))
		tmp = Float64(Float64(x - a) - Float64(a / z));
	else
		tmp = Float64(x - Float64(a * Float64(y / Float64(t + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -130000000.0) || ~((z <= 150.0)))
		tmp = (x - a) - (a / z);
	else
		tmp = x - (a * (y / (t + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -130000000.0], N[Not[LessEqual[z, 150.0]], $MachinePrecision]], N[(N[(x - a), $MachinePrecision] - N[(a / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(a * N[(y / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e8 or 150 < z

   1. Initial program 93.2%

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

      1. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 91.4%

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

   6. Taylor expanded in y around 0 65.3%

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

      1. cancel-sign-sub-inv 65.3%

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

      2. metadata-eval 65.3%

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

      3. *-lft-identity 65.3%

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

   8. Simplified 65.3%

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

   9. Taylor expanded in z around inf 80.0%

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

       1. associate-+r+ 80.0%

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

       2. mul-1-neg 80.0%

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

       3. unsub-neg 80.0%

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

       4. mul-1-neg 80.0%

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

       5. unsub-neg 80.0%

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

   11. Simplified 80.0%

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

   if -1.3e8 < z < 150

   1. Initial program 98.6%

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

      1. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 91.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000000 \lor \neg \left(z \leq 150\right):\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7700000 \lor \neg \left(z \leq 235\right):\\ \;\;\;\;x + \left(\frac{a}{\frac{z}{y}} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7700000.0) (not (<= z 235.0)))
   (+ x (- (/ a (/ z y)) a))
   (- x (* a (/ y (+ t 1.0))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7700000.0) || !(z <= 235.0)) {
		tmp = x + ((a / (z / y)) - a);
	} else {
		tmp = x - (a * (y / (t + 1.0)));
	}
	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 <= (-7700000.0d0)) .or. (.not. (z <= 235.0d0))) then
        tmp = x + ((a / (z / y)) - a)
    else
        tmp = x - (a * (y / (t + 1.0d0)))
    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 <= -7700000.0) || !(z <= 235.0)) {
		tmp = x + ((a / (z / y)) - a);
	} else {
		tmp = x - (a * (y / (t + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7700000.0) or not (z <= 235.0):
		tmp = x + ((a / (z / y)) - a)
	else:
		tmp = x - (a * (y / (t + 1.0)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7700000.0) || !(z <= 235.0))
		tmp = Float64(x + Float64(Float64(a / Float64(z / y)) - a));
	else
		tmp = Float64(x - Float64(a * Float64(y / Float64(t + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -7700000.0) || ~((z <= 235.0)))
		tmp = x + ((a / (z / y)) - a);
	else
		tmp = x - (a * (y / (t + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7700000.0], N[Not[LessEqual[z, 235.0]], $MachinePrecision]], N[(x + N[(N[(a / N[(z / y), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x - N[(a * N[(y / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.7e6 or 235 < z

   1. Initial program 93.3%

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

   3. Taylor expanded in z around inf 83.8%

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

      1. mul-1-neg 83.8%

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

      2. distribute-neg-frac 83.8%

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

   5. Simplified 83.8%

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

      1. frac-2neg 83.8%

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

      2. div-inv 83.8%

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

      3. sub-neg 83.8%

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

      4. distribute-neg-in 83.8%

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

      5. remove-double-neg 83.8%

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

      6. distribute-neg-frac 83.8%

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

      7. remove-double-neg 83.8%

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

      8. add-sqr-sqrt 42.9%

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

      9. sqrt-unprod 57.5%

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

      10. sqr-neg 57.5%

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

      11. sqrt-unprod 22.5%

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

      12. add-sqr-sqrt 44.9%

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

      13. clear-num 44.9%

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

      14. add-sqr-sqrt 22.5%

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

      15. sqrt-unprod 57.6%

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

      16. sqr-neg 57.6%

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

      17. sqrt-unprod 43.0%

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

      18. add-sqr-sqrt 83.8%

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

   7. Applied egg-rr 83.8%

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

      1. associate-*r/ 71.2%

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

      2. associate-/l* 83.8%

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

      3. +-commutative 83.8%

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

      4. unsub-neg 83.8%

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

   9. Simplified 83.8%

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

   10. Taylor expanded in z around 0 83.4%

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

       1. mul-1-neg 83.4%

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

       2. unsub-neg 83.4%

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

       3. associate-/l* 90.4%

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

   12. Simplified 90.4%

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

   if -7.7e6 < z < 235

   1. Initial program 98.6%

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

      1. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 91.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7700000 \lor \neg \left(z \leq 235\right):\\ \;\;\;\;x + \left(\frac{a}{\frac{z}{y}} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -37000000000 \lor \neg \left(z \leq 3.6 \cdot 10^{-22}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -37000000000.0) (not (<= z 3.6e-22))) (- x a) (- x (* y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -37000000000.0) || !(z <= 3.6e-22)) {
		tmp = x - a;
	} else {
		tmp = x - (y * 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 <= (-37000000000.0d0)) .or. (.not. (z <= 3.6d-22))) then
        tmp = x - a
    else
        tmp = x - (y * 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 <= -37000000000.0) || !(z <= 3.6e-22)) {
		tmp = x - a;
	} else {
		tmp = x - (y * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -37000000000.0) or not (z <= 3.6e-22):
		tmp = x - a
	else:
		tmp = x - (y * a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -37000000000.0) || !(z <= 3.6e-22))
		tmp = Float64(x - a);
	else
		tmp = Float64(x - Float64(y * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -37000000000.0) || ~((z <= 3.6e-22)))
		tmp = x - a;
	else
		tmp = x - (y * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -37000000000.0], N[Not[LessEqual[z, 3.6e-22]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(x - N[(y * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7e10 or 3.5999999999999998e-22 < z

   1. Initial program 92.8%

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

      1. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 79.3%

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

   if -3.7e10 < z < 3.5999999999999998e-22

   1. Initial program 99.3%

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

      1. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 73.4%

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

   6. Taylor expanded in z around 0 70.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -37000000000 \lor \neg \left(z \leq 3.6 \cdot 10^{-22}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -155000 \lor \neg \left(z \leq 3.4 \cdot 10^{-22}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -155000.0) (not (<= z 3.4e-22))) (- x a) x))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -155000.0) || !(z <= 3.4e-22)) {
		tmp = x - 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 ((z <= (-155000.0d0)) .or. (.not. (z <= 3.4d-22))) then
        tmp = x - 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 ((z <= -155000.0) || !(z <= 3.4e-22)) {
		tmp = x - a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -155000.0) or not (z <= 3.4e-22):
		tmp = x - a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -155000.0) || !(z <= 3.4e-22))
		tmp = Float64(x - a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -155000.0) || ~((z <= 3.4e-22)))
		tmp = x - a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -155000.0], N[Not[LessEqual[z, 3.4e-22]], $MachinePrecision]], N[(x - a), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -155000 or 3.3999999999999998e-22 < z

   1. Initial program 92.9%

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

      1. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 78.8%

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

   if -155000 < z < 3.3999999999999998e-22

   1. Initial program 99.3%

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

      1. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 73.2%

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

   6. Taylor expanded in x around inf 63.4%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -155000 \lor \neg \left(z \leq 3.4 \cdot 10^{-22}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.7% 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 96.1%

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

   1. associate-/r/ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in t around 0 81.8%

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

6. Taylor expanded in x around inf 56.8%

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

7. Final simplification 56.8%

   \[\leadsto x \]
8. Add Preprocessing

Developer target: 99.7% 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 2024027 
(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))))