Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 93.1% → 97.5%

Time: 8.0s

Alternatives: 9

Speedup: 1.0×

• 25.4% 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: 93.1% 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: 97.5% 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 92.8%

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

   1. associate-/l* 97.3%

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

3. Simplified 97.3%

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

   1. clear-num 97.2%

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

   2. un-div-inv 97.5%

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

6. Applied egg-rr 97.5%

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

7. Final simplification 97.5%

   \[\leadsto x + \frac{y - x}{\frac{t}{z}} \]
8. Add Preprocessing

Alternative 2: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+113} \lor \neg \left(x \leq 5.2 \cdot 10^{-57}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8.6e+113) (not (<= x 5.2e-57)))
   (* x (- 1.0 (/ z t)))
   (+ x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.6e+113) || !(x <= 5.2e-57)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y * (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-8.6d+113)) .or. (.not. (x <= 5.2d-57))) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = x + (y * (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.6e+113) || !(x <= 5.2e-57)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -8.6e+113) or not (x <= 5.2e-57):
		tmp = x * (1.0 - (z / t))
	else:
		tmp = x + (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -8.6e+113) || !(x <= 5.2e-57))
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(x + Float64(y * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -8.6e+113) || ~((x <= 5.2e-57)))
		tmp = x * (1.0 - (z / t));
	else
		tmp = x + (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -8.6e+113], N[Not[LessEqual[x, 5.2e-57]], $MachinePrecision]], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.6000000000000007e113 or 5.19999999999999971e-57 < x

   1. Initial program 90.5%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   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 94.2%

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

      1. mul-1-neg 94.2%

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

      2. unsub-neg 94.2%

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

   7. Simplified 94.2%

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

   if -8.6000000000000007e113 < x < 5.19999999999999971e-57

   1. Initial program 94.8%

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

      1. associate-/l* 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in y around inf 84.0%

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

      1. associate-*r/ 84.7%

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

   7. Simplified 84.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+113} \lor \neg \left(x \leq 5.2 \cdot 10^{-57}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{+115} \lor \neg \left(x \leq 1.05 \cdot 10^{-56}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.66e+115) (not (<= x 1.05e-56)))
   (* x (- 1.0 (/ z t)))
   (+ x (/ y (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.66e+115) || !(x <= 1.05e-56)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y / (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 <= (-1.66d+115)) .or. (.not. (x <= 1.05d-56))) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = x + (y / (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 <= -1.66e+115) || !(x <= 1.05e-56)) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.66e+115) or not (x <= 1.05e-56):
		tmp = x * (1.0 - (z / t))
	else:
		tmp = x + (y / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.66e+115) || !(x <= 1.05e-56))
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(x + Float64(y / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.66e+115) || ~((x <= 1.05e-56)))
		tmp = x * (1.0 - (z / t));
	else
		tmp = x + (y / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.66e+115], N[Not[LessEqual[x, 1.05e-56]], $MachinePrecision]], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.66e115 or 1.05000000000000003e-56 < x

   1. Initial program 90.5%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   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 94.2%

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

      1. mul-1-neg 94.2%

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

      2. unsub-neg 94.2%

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

   7. Simplified 94.2%

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

   if -1.66e115 < x < 1.05000000000000003e-56

   1. Initial program 94.8%

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

      1. associate-/l* 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in y around inf 84.0%

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

      1. associate-*r/ 84.7%

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

   7. Simplified 84.7%

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

      1. clear-num 84.7%

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

      2. div-inv 84.8%

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

   9. Applied egg-rr 84.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{+115} \lor \neg \left(x \leq 1.05 \cdot 10^{-56}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.1 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-57}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -9.1e+114)
   (* x (- 1.0 (/ z t)))
   (if (<= x 2.5e-57) (+ x (/ y (/ t z))) (- x (* x (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9.1e+114) {
		tmp = x * (1.0 - (z / t));
	} else if (x <= 2.5e-57) {
		tmp = x + (y / (t / z));
	} else {
		tmp = x - (x * (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-9.1d+114)) then
        tmp = x * (1.0d0 - (z / t))
    else if (x <= 2.5d-57) then
        tmp = x + (y / (t / z))
    else
        tmp = x - (x * (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9.1e+114) {
		tmp = x * (1.0 - (z / t));
	} else if (x <= 2.5e-57) {
		tmp = x + (y / (t / z));
	} else {
		tmp = x - (x * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -9.1e+114:
		tmp = x * (1.0 - (z / t))
	elif x <= 2.5e-57:
		tmp = x + (y / (t / z))
	else:
		tmp = x - (x * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -9.1e+114)
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	elseif (x <= 2.5e-57)
		tmp = Float64(x + Float64(y / Float64(t / z)));
	else
		tmp = Float64(x - Float64(x * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -9.1e+114)
		tmp = x * (1.0 - (z / t));
	elseif (x <= 2.5e-57)
		tmp = x + (y / (t / z));
	else
		tmp = x - (x * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -9.1e+114], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-57], N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.1000000000000001e114

   1. Initial program 81.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   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 94.4%

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

      1. mul-1-neg 94.4%

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

      2. unsub-neg 94.4%

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

   7. Simplified 94.4%

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

   if -9.1000000000000001e114 < x < 2.5000000000000001e-57

   1. Initial program 94.8%

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

      1. associate-/l* 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in y around inf 84.0%

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

      1. associate-*r/ 84.7%

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

   7. Simplified 84.7%

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

      1. clear-num 84.7%

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

      2. div-inv 84.8%

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

   9. Applied egg-rr 84.8%

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

   if 2.5000000000000001e-57 < x

   1. Initial program 94.2%

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

   3. Taylor expanded in y around 0 88.5%

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

      1. mul-1-neg 88.5%

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

      2. distribute-lft-neg-out 88.5%

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

      3. *-commutative 88.5%

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

   5. Simplified 88.5%

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

      1. frac-2neg 88.5%

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

      2. div-inv 88.5%

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

      3. distribute-rgt-neg-out 88.5%

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

      4. remove-double-neg 88.5%

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

      5. *-commutative 88.5%

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

   7. Applied egg-rr 88.5%

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

      1. *-commutative 88.5%

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

      2. add-sqr-sqrt 44.2%

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

      3. sqrt-unprod 57.8%

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

      4. sqr-neg 57.8%

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

      5. sqrt-unprod 21.7%

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

      6. add-sqr-sqrt 43.6%

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

      7. cancel-sign-sub 43.6%

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

      8. distribute-frac-neg2 43.6%

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

      9. *-commutative 43.6%

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

      10. associate-*l* 46.9%

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

      11. un-div-inv 46.9%

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

      12. add-sqr-sqrt 24.1%

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

      13. sqrt-unprod 60.8%

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

      14. sqr-neg 60.8%

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

      15. sqrt-unprod 46.5%

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

      16. add-sqr-sqrt 94.2%

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

   9. Applied egg-rr 94.2%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.1 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-57}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+49} \lor \neg \left(z \leq 4.8 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.8e+49) (not (<= z 4.8e+16))) (* x (/ z (- t))) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.8e+49) || !(z <= 4.8e+16)) {
		tmp = x * (z / -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 <= (-1.8d+49)) .or. (.not. (z <= 4.8d+16))) then
        tmp = x * (z / -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 <= -1.8e+49) || !(z <= 4.8e+16)) {
		tmp = x * (z / -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.8e+49) or not (z <= 4.8e+16):
		tmp = x * (z / -t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.8e+49) || !(z <= 4.8e+16))
		tmp = Float64(x * Float64(z / Float64(-t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.8e+49) || ~((z <= 4.8e+16)))
		tmp = x * (z / -t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.8e+49], N[Not[LessEqual[z, 4.8e+16]], $MachinePrecision]], N[(x * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.79999999999999998e49 or 4.8e16 < z

   1. Initial program 86.0%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   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 x around inf 63.3%

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

      1. mul-1-neg 63.3%

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

      2. unsub-neg 63.3%

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

   7. Simplified 63.3%

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

   8. Taylor expanded in z around inf 54.7%

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

      1. associate-*r/ 54.7%

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

      2. neg-mul-1 54.7%

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

   10. Simplified 54.7%

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

   if -1.79999999999999998e49 < z < 4.8e16

   1. Initial program 98.5%

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

      1. associate-/l* 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in z around 0 63.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+49} \lor \neg \left(z \leq 4.8 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z 1.55e-157) x (/ (* x z) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.55e-157) {
		tmp = x;
	} else {
		tmp = (x * z) / 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 (z <= 1.55d-157) then
        tmp = x
    else
        tmp = (x * z) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.55e-157) {
		tmp = x;
	} else {
		tmp = (x * z) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.55e-157:
		tmp = x
	else:
		tmp = (x * z) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.55e-157)
		tmp = x;
	else
		tmp = Float64(Float64(x * z) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.55e-157)
		tmp = x;
	else
		tmp = (x * z) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.55e-157], x, N[(N[(x * z), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.5499999999999999e-157

   1. Initial program 93.9%

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

      1. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in z around 0 48.9%

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

   if 1.5499999999999999e-157 < z

   1. Initial program 91.0%

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

   3. Taylor expanded in y around 0 55.8%

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

      1. mul-1-neg 55.8%

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

      2. distribute-lft-neg-out 55.8%

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

      3. *-commutative 55.8%

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

   5. Simplified 55.8%

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

   6. Taylor expanded in z around inf 57.8%

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

      1. mul-1-neg 57.8%

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

      2. +-commutative 57.8%

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

      3. unsub-neg 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in z around 0 21.7%

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

       1. associate-*r/ 30.1%

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

   11. Applied egg-rr 30.1%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% 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 92.8%

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

   1. associate-/l* 97.3%

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

3. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 8: 65.9% 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 92.8%

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

   1. associate-/l* 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in x around inf 69.7%

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

   1. mul-1-neg 69.7%

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

   2. unsub-neg 69.7%

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

7. Simplified 69.7%

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

8. Final simplification 69.7%

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

Alternative 9: 38.2% 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 92.8%

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

   1. associate-/l* 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in z around 0 39.2%

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

6. Final simplification 39.2%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 97.5% 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 2024078 
(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)))