Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 93.2% → 98.8%

Time: 7.0s

Alternatives: 10

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: 93.2% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+291}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* z (- y x)) t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+291)))
     (+ x (* z (/ (- y x) t)))
     t_1)))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + ((z * (y - x)) / t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+291)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + ((z * (y - x)) / t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e+291)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + ((z * (y - x)) / t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e+291):
		tmp = x + (z * ((y - x) / t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(z * Float64(y - x)) / t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+291))
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + ((z * (y - x)) / t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e+291)))
		tmp = x + (z * ((y - x) / t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+291]], $MachinePrecision]], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -inf.0 or 5.0000000000000001e291 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 77.8%

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

      1. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 5.0000000000000001e291

   1. Initial program 98.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{z \cdot \left(y - x\right)}{t} \leq -\infty \lor \neg \left(x + \frac{z \cdot \left(y - x\right)}{t} \leq 5 \cdot 10^{+291}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \]

Alternative 2: 97.7% 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 91.3%

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

   1. +-commutative 91.3%

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

   2. *-commutative 91.3%

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

   3. associate-*l/ 97.6%

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

   4. fma-def 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 3: 54.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -6.1 \cdot 10^{-112}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-219}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6e+48)
   x
   (if (<= t -6.1e-112)
     (/ (* z y) t)
     (if (<= t -5.6e-219)
       (* (/ z t) (- x))
       (if (<= t 1.7e+26) (* (/ z t) y) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6e+48) {
		tmp = x;
	} else if (t <= -6.1e-112) {
		tmp = (z * y) / t;
	} else if (t <= -5.6e-219) {
		tmp = (z / t) * -x;
	} else if (t <= 1.7e+26) {
		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 (t <= (-6d+48)) then
        tmp = x
    else if (t <= (-6.1d-112)) then
        tmp = (z * y) / t
    else if (t <= (-5.6d-219)) then
        tmp = (z / t) * -x
    else if (t <= 1.7d+26) 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 (t <= -6e+48) {
		tmp = x;
	} else if (t <= -6.1e-112) {
		tmp = (z * y) / t;
	} else if (t <= -5.6e-219) {
		tmp = (z / t) * -x;
	} else if (t <= 1.7e+26) {
		tmp = (z / t) * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6e+48:
		tmp = x
	elif t <= -6.1e-112:
		tmp = (z * y) / t
	elif t <= -5.6e-219:
		tmp = (z / t) * -x
	elif t <= 1.7e+26:
		tmp = (z / t) * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6e+48)
		tmp = x;
	elseif (t <= -6.1e-112)
		tmp = Float64(Float64(z * y) / t);
	elseif (t <= -5.6e-219)
		tmp = Float64(Float64(z / t) * Float64(-x));
	elseif (t <= 1.7e+26)
		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 (t <= -6e+48)
		tmp = x;
	elseif (t <= -6.1e-112)
		tmp = (z * y) / t;
	elseif (t <= -5.6e-219)
		tmp = (z / t) * -x;
	elseif (t <= 1.7e+26)
		tmp = (z / t) * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6e+48], x, If[LessEqual[t, -6.1e-112], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, -5.6e-219], N[(N[(z / t), $MachinePrecision] * (-x)), $MachinePrecision], If[LessEqual[t, 1.7e+26], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], x]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.9999999999999999e48 or 1.7000000000000001e26 < t

   1. Initial program 82.1%

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

   2. Taylor expanded in z around 0 68.5%

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

   if -5.9999999999999999e48 < t < -6.0999999999999999e-112

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 81.5%

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

   3. Taylor expanded in y around inf 59.2%

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

      1. *-commutative 59.2%

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

   5. Simplified 59.2%

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

   if -6.0999999999999999e-112 < t < -5.5999999999999998e-219

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 83.1%

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

   3. Taylor expanded in y around 0 58.8%

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

      1. associate-*r/ 58.8%

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

      2. mul-1-neg 58.8%

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

      3. distribute-rgt-neg-out 58.8%

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

      4. associate-*l/ 63.5%

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

   5. Simplified 63.5%

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

   if -5.5999999999999998e-219 < t < 1.7000000000000001e26

   1. Initial program 97.4%

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

   2. Taylor expanded in t around 0 80.4%

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

   3. Taylor expanded in y around inf 57.4%

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

      1. associate-*r/ 59.7%

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

   5. Simplified 59.7%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -6.1 \cdot 10^{-112}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-219}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-83} \lor \neg \left(z \leq 4.7 \cdot 10^{-126}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.3e-83) (not (<= z 4.7e-126))) (* z (/ (- y x) t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e-83) || !(z <= 4.7e-126)) {
		tmp = z * ((y - x) / 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 <= (-1.3d-83)) .or. (.not. (z <= 4.7d-126))) then
        tmp = z * ((y - x) / 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 <= -1.3e-83) || !(z <= 4.7e-126)) {
		tmp = z * ((y - x) / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.3e-83) or not (z <= 4.7e-126):
		tmp = z * ((y - x) / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.3e-83) || !(z <= 4.7e-126))
		tmp = Float64(z * Float64(Float64(y - x) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.3e-83) || ~((z <= 4.7e-126)))
		tmp = z * ((y - x) / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.3e-83], N[Not[LessEqual[z, 4.7e-126]], $MachinePrecision]], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.30000000000000004e-83 or 4.70000000000000017e-126 < z

   1. Initial program 88.3%

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

   2. Taylor expanded in t around 0 70.9%

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

      1. associate-*l/ 96.9%

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

   4. Applied egg-rr 75.0%

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

   if -1.30000000000000004e-83 < z < 4.70000000000000017e-126

   1. Initial program 97.3%

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

   2. Taylor expanded in z around 0 72.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-83} \lor \neg \left(z \leq 4.7 \cdot 10^{-126}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+157} \lor \neg \left(z \leq 8.2 \cdot 10^{+114}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.6e+157) (not (<= z 8.2e+114)))
   (* z (/ (- y x) t))
   (+ x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.6e+157) || !(z <= 8.2e+114)) {
		tmp = 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 ((z <= (-2.6d+157)) .or. (.not. (z <= 8.2d+114))) then
        tmp = 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 ((z <= -2.6e+157) || !(z <= 8.2e+114)) {
		tmp = z * ((y - x) / t);
	} else {
		tmp = x + ((z / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.6e+157) or not (z <= 8.2e+114):
		tmp = z * ((y - x) / t)
	else:
		tmp = x + ((z / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.6e+157) || !(z <= 8.2e+114))
		tmp = Float64(z * Float64(Float64(y - x) / t));
	else
		tmp = Float64(x + Float64(Float64(z / t) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.6e+157) || ~((z <= 8.2e+114)))
		tmp = z * ((y - x) / t);
	else
		tmp = x + ((z / t) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.6e+157], N[Not[LessEqual[z, 8.2e+114]], $MachinePrecision]], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.60000000000000011e157 or 8.2000000000000001e114 < z

   1. Initial program 85.3%

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

   2. Taylor expanded in t around 0 82.9%

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

      1. associate-*l/ 96.9%

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

   4. Applied egg-rr 91.6%

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

   if -2.60000000000000011e157 < z < 8.2000000000000001e114

   1. Initial program 93.9%

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

   2. Taylor expanded in y around inf 83.7%

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

      1. associate-*r/ 35.6%

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

   4. Simplified 87.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+157} \lor \neg \left(z \leq 8.2 \cdot 10^{+114}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \end{array} \]

Alternative 6: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-76} \lor \neg \left(y \leq 1.75 \cdot 10^{-107}\right):\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -9.6e-76) (not (<= y 1.75e-107)))
   (+ x (* (/ z t) y))
   (- x (* (/ z t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -9.6e-76) || !(y <= 1.75e-107)) {
		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 <= (-9.6d-76)) .or. (.not. (y <= 1.75d-107))) 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 <= -9.6e-76) || !(y <= 1.75e-107)) {
		tmp = x + ((z / t) * y);
	} else {
		tmp = x - ((z / t) * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -9.6e-76) or not (y <= 1.75e-107):
		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 <= -9.6e-76) || !(y <= 1.75e-107))
		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 <= -9.6e-76) || ~((y <= 1.75e-107)))
		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, -9.6e-76], N[Not[LessEqual[y, 1.75e-107]], $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 < -9.60000000000000053e-76 or 1.74999999999999993e-107 < y

   1. Initial program 90.8%

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

   2. Taylor expanded in y around inf 83.7%

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

      1. associate-*r/ 55.9%

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

   4. Simplified 90.3%

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

   if -9.60000000000000053e-76 < y < 1.74999999999999993e-107

   1. Initial program 92.1%

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

   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-lft-in 90.0%

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

      3. *-rgt-identity 90.0%

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

      4. mul-1-neg 90.0%

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

      5. distribute-rgt-neg-in 90.0%

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

      6. unsub-neg 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-76} \lor \neg \left(y \leq 1.75 \cdot 10^{-107}\right):\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot x\\ \end{array} \]

Alternative 7: 92.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{z}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -8.4e+40) (- x (* (/ z t) x)) (+ x (* z (/ (- y x) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -8.4e+40) {
		tmp = x - ((z / t) * x);
	} else {
		tmp = x + (z * ((y - x) / 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 <= (-8.4d+40)) then
        tmp = x - ((z / t) * x)
    else
        tmp = x + (z * ((y - x) / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -8.4e+40) {
		tmp = x - ((z / t) * x);
	} else {
		tmp = x + (z * ((y - x) / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -8.4e+40:
		tmp = x - ((z / t) * x)
	else:
		tmp = x + (z * ((y - x) / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -8.4e+40)
		tmp = Float64(x - Float64(Float64(z / t) * x));
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -8.4e+40)
		tmp = x - ((z / t) * x);
	else
		tmp = x + (z * ((y - x) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -8.4e+40], N[(x - N[(N[(z / t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.4000000000000004e40

   1. Initial program 86.1%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. *-commutative 98.5%

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

      2. distribute-lft-in 98.5%

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

      3. *-rgt-identity 98.5%

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

      4. mul-1-neg 98.5%

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

      5. distribute-rgt-neg-in 98.5%

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

      6. unsub-neg 98.5%

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

   4. Simplified 98.5%

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

   if -8.4000000000000004e40 < x

   1. Initial program 93.2%

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

      1. associate-*l/ 93.7%

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

   3. Applied egg-rr 93.7%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{z}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternative 8: 55.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+25}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.6e+39) x (if (<= t 6e+25) (* (/ z t) y) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.6e+39) {
		tmp = x;
	} else if (t <= 6e+25) {
		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 (t <= (-5.6d+39)) then
        tmp = x
    else if (t <= 6d+25) 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 (t <= -5.6e+39) {
		tmp = x;
	} else if (t <= 6e+25) {
		tmp = (z / t) * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.6e+39:
		tmp = x
	elif t <= 6e+25:
		tmp = (z / t) * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.6e+39)
		tmp = x;
	elseif (t <= 6e+25)
		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 (t <= -5.6e+39)
		tmp = x;
	elseif (t <= 6e+25)
		tmp = (z / t) * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.6e+39], x, If[LessEqual[t, 6e+25], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -5.60000000000000003e39 or 6.00000000000000011e25 < t

   1. Initial program 82.6%

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

   2. Taylor expanded in z around 0 67.6%

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

   if -5.60000000000000003e39 < t < 6.00000000000000011e25

   1. Initial program 98.3%

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

   2. Taylor expanded in t around 0 81.3%

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

   3. Taylor expanded in y around inf 55.3%

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

      1. associate-*r/ 56.8%

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

   5. Simplified 56.8%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+25}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 97.7% 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 91.3%

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

   1. associate-/l* 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 10: 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 91.3%

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

2. Taylor expanded in z around 0 40.4%

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

3. Final simplification 40.4%

   \[\leadsto x \]

Developer target: 98.0% 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 2023200 
(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)))