Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.6% → 97.7%

Time: 6.3s

Alternatives: 8

Speedup: 1.0×

• 25.2% 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: 92.6% 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: 97.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. +-commutative 94.7%

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

   2. associate-/l* 97.8%

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

   3. fma-define 97.8%

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

3. Simplified 97.8%

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

5. Final simplification 97.8%

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

Alternative 2: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+68} \lor \neg \left(y \leq 6.5 \cdot 10^{-83}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.6e+68) (not (<= y 6.5e-83)))
   (+ x (* y (/ z t)))
   (* x (- 1.0 (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.6e+68) || !(y <= 6.5e-83)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (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 ((y <= (-1.6d+68)) .or. (.not. (y <= 6.5d-83))) then
        tmp = x + (y * (z / t))
    else
        tmp = x * (1.0d0 - (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.6e+68) || !(y <= 6.5e-83)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.6e+68) or not (y <= 6.5e-83):
		tmp = x + (y * (z / t))
	else:
		tmp = x * (1.0 - (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.6e+68) || !(y <= 6.5e-83))
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.6e+68) || ~((y <= 6.5e-83)))
		tmp = x + (y * (z / t));
	else
		tmp = x * (1.0 - (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.6e+68], N[Not[LessEqual[y, 6.5e-83]], $MachinePrecision]], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.59999999999999997e68 or 6.5e-83 < y

   1. Initial program 92.4%

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

      1. associate-/l* 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in y around inf 87.2%

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

      1. associate-*r/ 92.0%

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

   7. Simplified 92.0%

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

   if -1.59999999999999997e68 < y < 6.5e-83

   1. Initial program 97.0%

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

      1. associate-/l* 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around inf 89.0%

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

      1. mul-1-neg 89.0%

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

      2. unsub-neg 89.0%

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

   7. Simplified 89.0%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+68} \lor \neg \left(y \leq 6.5 \cdot 10^{-83}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-56} \lor \neg \left(z \leq 1850000\right):\\ \;\;\;\;z \cdot \frac{x}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.8e-56) (not (<= z 1850000.0))) (* z (/ x (- t))) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.8e-56) || !(z <= 1850000.0)) {
		tmp = z * (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 <= (-6.8d-56)) .or. (.not. (z <= 1850000.0d0))) then
        tmp = z * (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 <= -6.8e-56) || !(z <= 1850000.0)) {
		tmp = z * (x / -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.8e-56) or not (z <= 1850000.0):
		tmp = z * (x / -t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.8e-56) || !(z <= 1850000.0))
		tmp = Float64(z * Float64(x / Float64(-t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.8e-56) || ~((z <= 1850000.0)))
		tmp = z * (x / -t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.8e-56], N[Not[LessEqual[z, 1850000.0]], $MachinePrecision]], N[(z * N[(x / (-t)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -6.79999999999999964e-56 or 1.85e6 < z

   1. Initial program 90.7%

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

      1. associate-/l* 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around inf 63.8%

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

      1. mul-1-neg 63.8%

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

      2. unsub-neg 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in t around 0 51.9%

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

      1. +-commutative 51.9%

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

      2. mul-1-neg 51.9%

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

      3. unsub-neg 51.9%

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

      4. *-commutative 51.9%

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

      5. distribute-lft-out-- 53.5%

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

   10. Simplified 53.5%

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

   11. Taylor expanded in t around 0 46.9%

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

       1. mul-1-neg 46.9%

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

       2. distribute-lft-neg-out 46.9%

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

       3. *-commutative 46.9%

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

   13. Simplified 46.9%

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

   14. Taylor expanded in z around 0 46.9%

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

       1. mul-1-neg 46.9%

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

       2. associate-*l/ 49.0%

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

       3. *-commutative 49.0%

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

       4. distribute-rgt-neg-in 49.0%

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

       5. distribute-neg-frac 49.0%

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

   16. Simplified 49.0%

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

   if -6.79999999999999964e-56 < z < 1.85e6

   1. Initial program 99.1%

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

      1. associate-/l* 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in z around 0 64.1%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-56} \lor \neg \left(z \leq 1850000\right):\\ \;\;\;\;z \cdot \frac{x}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-56}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;z \leq 16.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9e-56) (/ (* x (- z)) t) (if (<= z 16.5) x (* z (/ x (- t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9e-56) {
		tmp = (x * -z) / t;
	} else if (z <= 16.5) {
		tmp = x;
	} else {
		tmp = z * (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 (z <= (-9d-56)) then
        tmp = (x * -z) / t
    else if (z <= 16.5d0) then
        tmp = x
    else
        tmp = z * (x / -t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9e-56) {
		tmp = (x * -z) / t;
	} else if (z <= 16.5) {
		tmp = x;
	} else {
		tmp = z * (x / -t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9e-56:
		tmp = (x * -z) / t
	elif z <= 16.5:
		tmp = x
	else:
		tmp = z * (x / -t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9e-56)
		tmp = Float64(Float64(x * Float64(-z)) / t);
	elseif (z <= 16.5)
		tmp = x;
	else
		tmp = Float64(z * Float64(x / Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -9e-56)
		tmp = (x * -z) / t;
	elseif (z <= 16.5)
		tmp = x;
	else
		tmp = z * (x / -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9e-56], N[(N[(x * (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 16.5], x, N[(z * N[(x / (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.0000000000000001e-56

   1. Initial program 94.5%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 63.0%

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

      1. mul-1-neg 63.0%

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

      2. unsub-neg 63.0%

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

   7. Simplified 63.0%

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

   8. Taylor expanded in t around 0 49.7%

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

      1. +-commutative 49.7%

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

      2. mul-1-neg 49.7%

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

      3. unsub-neg 49.7%

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

      4. *-commutative 49.7%

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

      5. distribute-lft-out-- 52.6%

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

   10. Simplified 52.6%

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

   11. Taylor expanded in t around 0 48.5%

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

       1. mul-1-neg 48.5%

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

       2. distribute-lft-neg-out 48.5%

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

       3. *-commutative 48.5%

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

   13. Simplified 48.5%

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

   if -9.0000000000000001e-56 < z < 16.5

   1. Initial program 99.1%

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

      1. associate-/l* 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in z around 0 64.1%

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

   if 16.5 < z

   1. Initial program 86.2%

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

      1. associate-/l* 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in x around inf 64.7%

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

      1. mul-1-neg 64.7%

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

      2. unsub-neg 64.7%

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

   7. Simplified 64.7%

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

   8. Taylor expanded in t around 0 54.3%

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

      1. +-commutative 54.3%

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

      2. mul-1-neg 54.3%

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

      3. unsub-neg 54.3%

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

      4. *-commutative 54.3%

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

      5. distribute-lft-out-- 54.5%

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

   10. Simplified 54.5%

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

   11. Taylor expanded in t around 0 45.1%

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

       1. mul-1-neg 45.1%

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

       2. distribute-lft-neg-out 45.1%

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

       3. *-commutative 45.1%

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

   13. Simplified 45.1%

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

   14. Taylor expanded in z around 0 45.1%

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

       1. mul-1-neg 45.1%

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

       2. associate-*l/ 49.6%

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

       3. *-commutative 49.6%

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

       4. distribute-rgt-neg-in 49.6%

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

       5. distribute-neg-frac 49.6%

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

   16. Simplified 49.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-56}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;z \leq 16.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+64}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= x 1e+64) x (* t (/ x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1e+64) {
		tmp = x;
	} else {
		tmp = t * (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 <= 1d+64) then
        tmp = x
    else
        tmp = t * (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 <= 1e+64) {
		tmp = x;
	} else {
		tmp = t * (x / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1e+64:
		tmp = x
	else:
		tmp = t * (x / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1e+64)
		tmp = x;
	else
		tmp = Float64(t * Float64(x / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1e+64)
		tmp = x;
	else
		tmp = t * (x / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1e+64], x, N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000002e64

   1. Initial program 94.9%

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

      1. associate-/l* 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in z around 0 37.5%

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

   if 1.00000000000000002e64 < x

   1. Initial program 93.7%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   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 84.6%

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

      1. mul-1-neg 84.6%

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

      2. unsub-neg 84.6%

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

   7. Simplified 84.6%

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

   8. Taylor expanded in t around 0 63.7%

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

      1. +-commutative 63.7%

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

      2. mul-1-neg 63.7%

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

      3. unsub-neg 63.7%

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

      4. *-commutative 63.7%

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

      5. distribute-lft-out-- 66.1%

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

   10. Simplified 66.1%

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

   11. Taylor expanded in t around inf 27.3%

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

       1. *-commutative 27.3%

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

   13. Simplified 27.3%

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

       1. *-commutative 27.3%

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

       2. associate-/l* 50.4%

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

   15. Applied egg-rr 50.4%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+64}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% 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 94.7%

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

   1. associate-/l* 97.8%

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

3. Simplified 97.8%

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

5. Final simplification 97.8%

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

Alternative 7: 65.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x * (1.0 - (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 * (1.0d0 - (z / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. associate-/l* 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in x around inf 68.7%

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

   1. mul-1-neg 68.7%

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

   2. unsub-neg 68.7%

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

7. Simplified 68.7%

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

8. Final simplification 68.7%

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

Alternative 8: 38.7% 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 94.7%

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

   1. associate-/l* 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in z around 0 37.5%

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

6. Final simplification 37.5%

   \[\leadsto x \]
7. 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 2024059 
(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)))