Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.8% → 97.8%

Time: 7.7s

Alternatives: 8

Speedup: 1.0×

• 24.5% 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: 92.8% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

   1. +-commutative 92.4%

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

   2. *-commutative 92.4%

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

   3. associate-*l/ 98.5%

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

   4. fma-def 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+218}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+56}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.1e+218)
   (+ x (* (/ z t) y))
   (if (<= t 1.3e+56) (+ x (/ (* z (- y x)) t)) (+ x (/ z (/ t y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.1e+218) {
		tmp = x + ((z / t) * y);
	} else if (t <= 1.3e+56) {
		tmp = x + ((z * (y - x)) / t);
	} else {
		tmp = x + (z / (t / y));
	}
	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 (t <= (-3.1d+218)) then
        tmp = x + ((z / t) * y)
    else if (t <= 1.3d+56) then
        tmp = x + ((z * (y - x)) / t)
    else
        tmp = x + (z / (t / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.1e+218) {
		tmp = x + ((z / t) * y);
	} else if (t <= 1.3e+56) {
		tmp = x + ((z * (y - x)) / t);
	} else {
		tmp = x + (z / (t / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.1e+218:
		tmp = x + ((z / t) * y)
	elif t <= 1.3e+56:
		tmp = x + ((z * (y - x)) / t)
	else:
		tmp = x + (z / (t / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.1e+218)
		tmp = Float64(x + Float64(Float64(z / t) * y));
	elseif (t <= 1.3e+56)
		tmp = Float64(x + Float64(Float64(z * Float64(y - x)) / t));
	else
		tmp = Float64(x + Float64(z / Float64(t / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.1e+218)
		tmp = x + ((z / t) * y);
	elseif (t <= 1.3e+56)
		tmp = x + ((z * (y - x)) / t);
	else
		tmp = x + (z / (t / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.1e+218], N[(x + N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e+56], N[(x + N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.1000000000000002e218

   1. Initial program 77.0%

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

   2. Taylor expanded in y around inf 86.7%

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

      1. associate-*r/ 100.0%

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

   4. Simplified 100.0%

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

   if -3.1000000000000002e218 < t < 1.30000000000000005e56

   1. Initial program 97.4%

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

   if 1.30000000000000005e56 < t

   1. Initial program 79.1%

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

   2. Taylor expanded in y around inf 83.1%

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

      1. associate-*r/ 89.9%

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

   4. Simplified 89.9%

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

   5. Taylor expanded in y around 0 83.1%

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

      1. *-commutative 83.1%

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

      2. associate-/l* 90.7%

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

   7. Simplified 90.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+218}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+56}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \end{array} \]

Alternative 3: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-115} \lor \neg \left(y \leq 2.75 \cdot 10^{-73}\right):\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.65e-115) (not (<= y 2.75e-73)))
   (+ x (* (/ z t) y))
   (- x (* (/ z t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.65e-115) || !(y <= 2.75e-73)) {
		tmp = x + ((z / t) * y);
	} else {
		tmp = x - ((z / t) * 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 ((y <= (-1.65d-115)) .or. (.not. (y <= 2.75d-73))) then
        tmp = x + ((z / t) * y)
    else
        tmp = x - ((z / t) * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.65e-115) || !(y <= 2.75e-73)) {
		tmp = x + ((z / t) * y);
	} else {
		tmp = x - ((z / t) * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.65e-115) or not (y <= 2.75e-73):
		tmp = x + ((z / t) * y)
	else:
		tmp = x - ((z / t) * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.65e-115) || !(y <= 2.75e-73))
		tmp = Float64(x + Float64(Float64(z / t) * y));
	else
		tmp = Float64(x - Float64(Float64(z / t) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.65e-115) || ~((y <= 2.75e-73)))
		tmp = x + ((z / t) * y);
	else
		tmp = x - ((z / t) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.65e-115], N[Not[LessEqual[y, 2.75e-73]], $MachinePrecision]], N[(x + N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z / t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.64999999999999995e-115 or 2.75000000000000003e-73 < y

   1. Initial program 90.9%

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

   2. Taylor expanded in y around inf 85.6%

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

      1. associate-*r/ 90.2%

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

   4. Simplified 90.2%

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

   if -1.64999999999999995e-115 < y < 2.75000000000000003e-73

   1. Initial program 95.4%

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

   2. Taylor expanded in x around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. distribute-rgt-in 90.0%

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

      3. *-lft-identity 90.0%

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

      4. mul-1-neg 90.0%

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

      5. cancel-sign-sub-inv 90.0%

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

      6. *-commutative 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-115} \lor \neg \left(y \leq 2.75 \cdot 10^{-73}\right):\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot x\\ \end{array} \]

Alternative 4: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8000000 \lor \neg \left(z \leq 4.7 \cdot 10^{-57}\right):\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8000000.0) (not (<= z 4.7e-57))) (* z (/ y t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8000000.0) || !(z <= 4.7e-57)) {
		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 <= (-8000000.0d0)) .or. (.not. (z <= 4.7d-57))) 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 <= -8000000.0) || !(z <= 4.7e-57)) {
		tmp = z * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8000000.0) or not (z <= 4.7e-57):
		tmp = z * (y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8000000.0) || !(z <= 4.7e-57))
		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 <= -8000000.0) || ~((z <= 4.7e-57)))
		tmp = z * (y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8000000.0], N[Not[LessEqual[z, 4.7e-57]], $MachinePrecision]], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -8e6 or 4.6999999999999998e-57 < z

   1. Initial program 86.4%

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

      1. associate-/l* 97.7%

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

      2. div-sub 88.0%

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

      3. associate-+r- 88.0%

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

      4. div-inv 88.0%

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

      5. clear-num 88.0%

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

      6. div-inv 88.0%

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

      7. clear-num 88.0%

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

   3. Applied egg-rr 88.0%

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

   4. Taylor expanded in t around 0 70.9%

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

   5. Taylor expanded in y around inf 47.9%

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

      1. *-commutative 47.9%

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

   7. Simplified 47.9%

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

      1. associate-/l* 52.8%

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

      2. clear-num 52.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{t}{y}}{z}}} \]

      3. associate-/r/ 52.8%

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

      4. clear-num 53.1%

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

   9. Applied egg-rr 53.1%

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

   if -8e6 < z < 4.6999999999999998e-57

   1. Initial program 98.4%

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

   2. Taylor expanded in z around 0 63.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8000000 \lor \neg \left(z \leq 4.7 \cdot 10^{-57}\right):\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 53.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+37} \lor \neg \left(y \leq 3.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.2e+37) (not (<= y 3.2e-9))) (* (/ z t) y) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.2e+37) || !(y <= 3.2e-9)) {
		tmp = (z / t) * y;
	} 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 ((y <= (-2.2d+37)) .or. (.not. (y <= 3.2d-9))) then
        tmp = (z / t) * y
    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 ((y <= -2.2e+37) || !(y <= 3.2e-9)) {
		tmp = (z / t) * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.2e+37) or not (y <= 3.2e-9):
		tmp = (z / t) * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.2e+37) || !(y <= 3.2e-9))
		tmp = Float64(Float64(z / t) * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -2.2e+37) || ~((y <= 3.2e-9)))
		tmp = (z / t) * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.2e+37], N[Not[LessEqual[y, 3.2e-9]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.2000000000000001e37 or 3.20000000000000012e-9 < y

   1. Initial program 91.6%

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

      1. associate-/l* 99.1%

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

      2. div-sub 93.5%

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

      3. associate-+r- 93.5%

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

      4. div-inv 93.5%

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

      5. clear-num 93.8%

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

      6. div-inv 93.8%

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

      7. clear-num 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in t around 0 62.4%

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

   5. Taylor expanded in y around inf 58.7%

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

      1. *-commutative 58.7%

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

   7. Simplified 58.7%

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

      1. associate-*l/ 64.1%

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

   9. Applied egg-rr 64.1%

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

   if -2.2000000000000001e37 < y < 3.20000000000000012e-9

   1. Initial program 93.2%

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

   2. Taylor expanded in z around 0 55.0%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+37} \lor \neg \left(y \leq 3.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y - x}{\frac{t}{z}} \end{array} \]

FPCore

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

C

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

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) / (t / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

   1. associate-/l* 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

   \[\leadsto x + \frac{y - x}{\frac{t}{z}} \]

Alternative 7: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + \frac{z}{t} \cdot y \end{array} \]

FPCore

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

C

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

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 + ((z / t) * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

2. Taylor expanded in y around inf 76.8%

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

   1. associate-*r/ 80.6%

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

4. Simplified 80.6%

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

5. Final simplification 80.6%

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

Alternative 8: 39.6% 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 92.4%

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

2. Taylor expanded in z around 0 42.7%

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

3. Final simplification 42.7%

   \[\leadsto x \]

Developer target: 97.8% accurate, 0.7× 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 2023228 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (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)))