Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 93.3% → 97.8%

Time: 7.9s

Alternatives: 7

Speedup: 1.0×

• 25.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. associate-/l* 97.2%

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

3. Simplified 97.2%

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

Alternative 2: 52.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{t}\\ \mathbf{if}\;t \leq -2.35 \cdot 10^{+83}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-172}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+152}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) t)))
   (if (<= t -2.35e+83)
     x
     (if (<= t -9e-172)
       t_1
       (if (<= t 1.9e-216) (* x (/ (- z) t)) (if (<= t 2.9e+152) t_1 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / t;
	double tmp;
	if (t <= -2.35e+83) {
		tmp = x;
	} else if (t <= -9e-172) {
		tmp = t_1;
	} else if (t <= 1.9e-216) {
		tmp = x * (-z / t);
	} else if (t <= 2.9e+152) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * z) / t
    if (t <= (-2.35d+83)) then
        tmp = x
    else if (t <= (-9d-172)) then
        tmp = t_1
    else if (t <= 1.9d-216) then
        tmp = x * (-z / t)
    else if (t <= 2.9d+152) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / t;
	double tmp;
	if (t <= -2.35e+83) {
		tmp = x;
	} else if (t <= -9e-172) {
		tmp = t_1;
	} else if (t <= 1.9e-216) {
		tmp = x * (-z / t);
	} else if (t <= 2.9e+152) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) / t
	tmp = 0
	if t <= -2.35e+83:
		tmp = x
	elif t <= -9e-172:
		tmp = t_1
	elif t <= 1.9e-216:
		tmp = x * (-z / t)
	elif t <= 2.9e+152:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / t)
	tmp = 0.0
	if (t <= -2.35e+83)
		tmp = x;
	elseif (t <= -9e-172)
		tmp = t_1;
	elseif (t <= 1.9e-216)
		tmp = Float64(x * Float64(Float64(-z) / t));
	elseif (t <= 2.9e+152)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) / t;
	tmp = 0.0;
	if (t <= -2.35e+83)
		tmp = x;
	elseif (t <= -9e-172)
		tmp = t_1;
	elseif (t <= 1.9e-216)
		tmp = x * (-z / t);
	elseif (t <= 2.9e+152)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -2.35e+83], x, If[LessEqual[t, -9e-172], t$95$1, If[LessEqual[t, 1.9e-216], N[(x * N[((-z) / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.9e+152], t$95$1, x]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.3499999999999999e83 or 2.8999999999999998e152 < t

   1. Initial program 83.7%

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

      1. associate-/l* 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in z around 0 73.3%

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

   if -2.3499999999999999e83 < t < -9.00000000000000008e-172 or 1.9e-216 < t < 2.8999999999999998e152

   1. Initial program 98.5%

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

      1. associate-/l* 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in x around 0 92.3%

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

   6. Taylor expanded in x around 0 57.4%

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

   if -9.00000000000000008e-172 < t < 1.9e-216

   1. Initial program 94.9%

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

   3. Taylor expanded in y around 0 70.4%

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

      1. mul-1-neg 70.4%

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

      2. distribute-lft-neg-out 70.4%

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

      3. *-commutative 70.4%

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

   5. Simplified 70.4%

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

      1. div-inv 70.4%

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

      2. *-commutative 70.4%

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

      3. add-sqr-sqrt 31.8%

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

      4. sqrt-unprod 39.6%

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

      5. sqr-neg 39.6%

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

      6. sqrt-unprod 11.1%

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

      7. add-sqr-sqrt 13.2%

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

      8. *-commutative 13.2%

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

      9. remove-double-neg 13.2%

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

      10. distribute-rgt-neg-out 13.2%

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

      11. cancel-sign-sub-inv 13.2%

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

      12. associate-*l* 13.0%

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

      13. div-inv 13.0%

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

      14. add-sqr-sqrt 2.2%

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

      15. sqrt-unprod 23.8%

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

      16. sqr-neg 23.8%

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

      17. sqrt-unprod 28.9%

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

      18. add-sqr-sqrt 55.7%

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

   7. Applied egg-rr 55.7%

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

      1. clear-num 55.7%

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

      2. un-div-inv 57.8%

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

   9. Applied egg-rr 57.8%

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

   10. Taylor expanded in z around inf 60.3%

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

       1. mul-1-neg 60.3%

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

       2. associate-*r/ 60.8%

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

       3. *-commutative 60.8%

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

       4. distribute-rgt-neg-in 60.8%

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

   12. Simplified 60.8%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{+83}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-172}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+152}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-15} \lor \neg \left(x \leq 2.2 \cdot 10^{+91}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.65e-15) (not (<= x 2.2e+91)))
   (* x (- 1.0 (/ z t)))
   (+ x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.65e-15) || !(x <= 2.2e+91)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y * (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.65d-15)) .or. (.not. (x <= 2.2d+91))) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = x + (y * (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.65e-15) || !(x <= 2.2e+91)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.65e-15) or not (x <= 2.2e+91):
		tmp = x * (1.0 - (z / t))
	else:
		tmp = x + (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.65e-15) || !(x <= 2.2e+91))
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(x + Float64(y * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.65e-15) || ~((x <= 2.2e+91)))
		tmp = x * (1.0 - (z / t));
	else
		tmp = x + (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.65e-15], N[Not[LessEqual[x, 2.2e+91]], $MachinePrecision]], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e-15 or 2.19999999999999999e91 < x

   1. Initial program 90.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 88.1%

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

      1. mul-1-neg 88.1%

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

      2. unsub-neg 88.1%

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

   7. Simplified 88.1%

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

   if -1.65e-15 < x < 2.19999999999999999e91

   1. Initial program 95.7%

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

      1. associate-/l* 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in y around inf 86.8%

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

      1. associate-*r/ 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-15} \lor \neg \left(x \leq 2.2 \cdot 10^{+91}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-226} \lor \neg \left(x \leq 1.3 \cdot 10^{-115}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.8e-226) (not (<= x 1.3e-115)))
   (* x (- 1.0 (/ z t)))
   (/ (* y z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.8e-226) || !(x <= 1.3e-115)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = (y * z) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-4.8d-226)) .or. (.not. (x <= 1.3d-115))) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = (y * z) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.8e-226) || !(x <= 1.3e-115)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = (y * z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.8e-226) or not (x <= 1.3e-115):
		tmp = x * (1.0 - (z / t))
	else:
		tmp = (y * z) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.8e-226) || !(x <= 1.3e-115))
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(Float64(y * z) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.8e-226) || ~((x <= 1.3e-115)))
		tmp = x * (1.0 - (z / t));
	else
		tmp = (y * z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.8e-226], N[Not[LessEqual[x, 1.3e-115]], $MachinePrecision]], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7999999999999999e-226 or 1.30000000000000002e-115 < x

   1. Initial program 93.2%

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

      1. associate-/l* 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. unsub-neg 75.9%

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

   7. Simplified 75.9%

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

   if -4.7999999999999999e-226 < x < 1.30000000000000002e-115

   1. Initial program 94.6%

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

      1. associate-/l* 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in x around 0 89.2%

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

   6. Taylor expanded in x around 0 76.5%

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-226} \lor \neg \left(x \leq 1.3 \cdot 10^{-115}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-61} \lor \neg \left(z \leq 1.25 \cdot 10^{-204}\right):\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.06e-61) (not (<= z 1.25e-204))) (* z (/ y t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.06e-61) || !(z <= 1.25e-204)) {
		tmp = z * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.06d-61)) .or. (.not. (z <= 1.25d-204))) then
        tmp = z * (y / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.06e-61) || !(z <= 1.25e-204)) {
		tmp = z * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.06e-61) or not (z <= 1.25e-204):
		tmp = z * (y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.06e-61) || !(z <= 1.25e-204))
		tmp = Float64(z * Float64(y / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.06e-61) || ~((z <= 1.25e-204)))
		tmp = z * (y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.06e-61], N[Not[LessEqual[z, 1.25e-204]], $MachinePrecision]], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.0599999999999999e-61 or 1.25e-204 < z

   1. Initial program 91.2%

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

      1. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in x around 0 85.8%

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

   6. Taylor expanded in x around 0 55.8%

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

      1. associate-*l/ 55.2%

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

      2. *-commutative 55.2%

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

   8. Simplified 55.2%

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

   if -1.0599999999999999e-61 < z < 1.25e-204

   1. Initial program 98.3%

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

      1. associate-/l* 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around 0 63.1%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-61} \lor \neg \left(z \leq 1.25 \cdot 10^{-204}\right):\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-61}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-204}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2e-61) (* z (/ y t)) (if (<= z 1.25e-204) x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e-61) {
		tmp = z * (y / t);
	} else if (z <= 1.25e-204) {
		tmp = x;
	} else {
		tmp = (y * z) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2d-61)) then
        tmp = z * (y / t)
    else if (z <= 1.25d-204) then
        tmp = x
    else
        tmp = (y * z) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e-61) {
		tmp = z * (y / t);
	} else if (z <= 1.25e-204) {
		tmp = x;
	} else {
		tmp = (y * z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2e-61:
		tmp = z * (y / t)
	elif z <= 1.25e-204:
		tmp = x
	else:
		tmp = (y * z) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2e-61)
		tmp = Float64(z * Float64(y / t));
	elseif (z <= 1.25e-204)
		tmp = x;
	else
		tmp = Float64(Float64(y * z) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2e-61)
		tmp = z * (y / t);
	elseif (z <= 1.25e-204)
		tmp = x;
	else
		tmp = (y * z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2e-61], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e-204], x, N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.0000000000000001e-61

   1. Initial program 93.5%

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

      1. associate-/l* 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 85.3%

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

   6. Taylor expanded in x around 0 56.6%

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

      1. associate-*l/ 59.1%

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

      2. *-commutative 59.1%

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

   8. Simplified 59.1%

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

   if -2.0000000000000001e-61 < z < 1.25e-204

   1. Initial program 98.3%

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

      1. associate-/l* 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around 0 63.1%

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

   if 1.25e-204 < z

   1. Initial program 89.5%

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

      1. associate-/l* 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around 0 86.1%

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

   6. Taylor expanded in x around 0 55.2%

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 38.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 x)

C

double code(double x, double y, double z, double t) {
	return x;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

public static double code(double x, double y, double z, double t) {
	return x;
}

Python

def code(x, y, z, t):
	return x

Julia

function code(x, y, z, t)
	return x
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 93.6%

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

   1. associate-/l* 97.2%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 36.5%

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

Developer target: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< x -9.025511195533005e-135)
   (- x (* (/ z t) (- x y)))
   (if (< x 4.275032163700715e-250)
     (+ x (* (/ (- y x) t) z))
     (+ x (/ (- y x) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x < (-9.025511195533005d-135)) then
        tmp = x - ((z / t) * (x - y))
    else if (x < 4.275032163700715d-250) then
        tmp = x + (((y - x) / t) * z)
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x < -9.025511195533005e-135:
		tmp = x - ((z / t) * (x - y))
	elif x < 4.275032163700715e-250:
		tmp = x + (((y - x) / t) * z)
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x < -9.025511195533005e-135)
		tmp = Float64(x - Float64(Float64(z / t) * Float64(x - y)));
	elseif (x < 4.275032163700715e-250)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x < -9.025511195533005e-135)
		tmp = x - ((z / t) * (x - y));
	elseif (x < 4.275032163700715e-250)
		tmp = x + (((y - x) / t) * z);
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[x, -9.025511195533005e-135], N[(x - N[(N[(z / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[x, 4.275032163700715e-250], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024091 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :alt
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

  (+ x (/ (* (- y x) z) t)))